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Riemann–Finsler GeometryMSRI PublicationsVolume 50, 2004

Volumes on Normed and Finsler Spaces

J. C. ALVAREZ PAIVA AND A. C. THOMPSON

Contents

1. Introduction 12. A Short Review of the Geometry of Normed Spaces 3

2.1. Maps between normed spaces 52.2. Dual spaces and polar bodies 62.3. Sociology of normed spaces 7

3. Volumes on Normed Spaces 93.1. Examples of definitions of volume in normed spaces 103.2. The volume of the unit ball 133.3. Relationship between the definitions of volume 143.4. Extension to Finsler manifolds 15

4. k-Volume Densities 174.1. Examples of k-volume densities 184.2. Convexity of (n−1)-volume densities 204.3. Convexity properties of k-volume densities 26

5. Length and Area in Two-Dimensional Normed Spaces 306. Area on Finite-Dimensional Normed Spaces 35

6.1. Injectivity and range of the area definition 366.2. Area of the unit sphere 396.3. Mixed volumes and the isoperimetrix 406.4. Geometry of the isoperimetrix 43

Acknowledgments 45References 45

1. Introduction

The study of volumes and areas on normed and Finsler spaces is a relatively

new field that comprises and unifies large domains of convexity, geometric to-

mography, and integral geometry. It opens many classical unsolved problems in

these fields to powerful techniques in global differential geometry, and suggests

new challenging problems that are delightfully geometric and simple to state.

Keywords: Minkowski geometry, Hausdorff measure, Holmes–Thompson volume, Finsler man-

ifold, isoperimetric inequality.

1

2 J. C. ALVAREZ PAIVA AND A. C. THOMPSON

The theory starts with a simple question: How does one measure volume

on a finite-dimensional normed space? At first sight, this question may seem

either unmotivated or trivial: normed spaces are metric spaces and we can mea-

sure volume using the Hausdorff measure, period. However, if one starts asking

simple, naive questions one discovers the depth of the problem. Even if one is

unwilling to consider that definitions of volume other than the Hausdorff mea-

sure are not only possible but may even be better, one is faced with questions

such as these: What is the (n−1)-dimensional Hausdorff measure of the unit

sphere of an n-dimensional normed space? Do flat regions minimize area? For

what normed spaces are metric balls also the solutions of the isoperimetric prob-

lem? These questions, first posed in convex-geometric language by Busemann

and Petty [1956], are still open, at least in their full generality. However, one

should not assume too quickly that the subject is impossible. Some beautiful

results and striking connections have been found. For example, the fact that the

(n−1)-Hausdorff measure in a normed space determines the norm is equivalent

to the fact that the areas of the central sections determine a convex body that

is symmetric with respect to the origin. This, in turn, follows from the study of

the spherical Radon transform. The fact that regions in hyperplanes are area-

minimizing is equivalent to the fact that the intersection body of a convex body

that is symmetric with respect to the origin is also convex.

But the true interest of the theory can only be appreciated if one is willing

to challenge Busemann’s dictum that the natural volume in a normed or Finsler

space is the Hausdorff measure. Indeed, thinking of a normed or Finsler space as

an anisotropic medium where the speed of a light ray depends on its direction,

we are led to consider a completely different notion of volume, which has become

known as the Holmes–Thompson volume. This notion of volume, introduced in

[Holmes and Thompson 1979], uncovers striking connections between integral

geometry, convexity, and Hamiltonian systems. For example, in a recent series

of papers, [Schneider and Wieacker 1997], [Alvarez and Fernandes 1998], [Alvarez

and Fernandes 1999], [Schneider 2001], and [Schneider 2002], it was shown that

the classical integral geometric formulas in Euclidean spaces can be extended to

normed and even to projective Finsler spaces (the solutions of Hilbert’s fourth

problem) if the areas of submanifolds are measured with the Holmes–Thompson

definition. That extensions of this kind are not possible with the Busemann

definition was shown by Schneider [Schneider 2001].

Using Finsler techniques, Burago and Ivanov [2001] have proved that a flat

two-dimensional disc in a finite-dimensional normed space minimizes area among

all other immersed discs with the same boundary. Ivanov [2001] has shown,

among other things, that Pu’s isosystolic inequality for Riemannian metrics in

the projective plane extends to Finsler metrics, and the Finslerian extension

of Berger’s infinitesimal isosystolic inequality for Riemannian metrics on real

projective spaces of arbitrary dimension has been proved by Alvarez [2002].

VOLUMES ON NORMED AND FINSLER SPACES 3

Despite these and other recent interdisciplinary results, we believe that the

most surprising feature of the Holmes–Thompson definition is the way in which

it organizes a large portion of convexity into a coherent theory. For example,

the sharp upper bound for the volume of the unit ball of a normed space is

given by the Blaschke–Santalo inequality; the conjectured sharp lower bound is

Mahler’s conjecture; and the reconstruction of the norm from the area functional

is equivalent to the famous Minkowski’s problem of reconstructing a convex body

from the knowledge of its curvature as a function of its unit normals.

In this paper, we have attempted to provide students and researchers in Finsler

and global differential geometry with a clear and concise introduction to the

theory of volumes on normed and Finsler spaces. To do this, we have avoided

the temptation to use auxiliary Euclidean structures to describe the various

concepts and constructions. While these auxiliary structures may render some

of the proofs simpler, they hinder the understanding of the subject and make the

application of the ideas and techniques to Finsler spaces much more cumbersome.

We also believe that by presenting the results and techniques in intrinsic terms we

can make some of the beautiful results of convexity more accessible and enticing

to differential geometers.

In the course of our writing we had to make some tough choices as to what

material should be left out as either too advanced or too specialized. At the

end we decided that we would concentrate on the basic questions and techniques

of the theory, while doing our best to exhibit the general abstract framework

that makes the theory of volumes on normed spaces into a sort of Grand Unified

Theory for many problems in convexity and Finsler geometry. As a result there

is little Finsler geometry per se in the pages that follow. However, just as

tensors, forms, spinors, and Clifford algebras developed in invariant form have

immediate use in Riemannian geometry, the more geometric constructions with

norms, convex bodies, and k-volume densities that make up the heart of this

paper have immediate applications to Finsler geometry.

2. A Short Review of the Geometry of Normed Spaces

This section is a quick review of the geometry of finite-dimensional normed

spaces adapted to the needs and language of Finsler geometry. Unless stated

otherwise, all vector spaces in this article are real and finite-dimensional. We

suggest that the reader merely browse through this section and come back to it

if and when it becomes necessary.

Definition 2.1. A norm on a vector space X is a function

‖ · ‖ : X → [0,∞)

satisfying the following properties of positivity, homogeneity, and convexity:

(1) If ‖x‖ = 0, then x = 0;


(2) If t is a real number, then ‖tx‖ = |t|‖x‖;

(3) For any two vectors x and y in X, ‖x + y‖ ≤ ‖x‖ + ‖y‖.

If (X, ‖ · ‖) is a finite-dimensional normed space, the set

BX := x ∈ X : ‖x‖ ≤ 1

is the unit ball of X and the boundary of BX , SX , is its unit sphere. Notice that

BX is a compact, convex set with nonempty interior. In short, it is a convex

body in X. Moreover, it is symmetric with respect to the origin. Conversely, if

B ⊂ X is a centered convex body (i.e., a convex body symmetric with respect to

the origin), it is the unit ball of the norm

‖x‖ := inf t ≥ 0 : ty = x for some y ∈ B.

We shall now describe various classes of normed spaces that will appear re-

peatedly throughout the paper.

Euclidean spaces. A Euclidean structure on a finite-dimensional vector space X

is a choice of a symmetric, positive-definite quadratic form Q : X → R. The

normed space (X,Q1/2) will be called a Euclidean space. Note that a normed

space is Euclidean if and only if its unit sphere is an ellipsoid, which is if and

only if the norm satisfies the parallelogram identity:

‖x + y‖2 + ‖x − y‖2 = 2‖x‖2 + 2‖y‖2.

Exercise 2.2. Let B ⊂ Rn be a convex body symmetric with respect to the

origin. Show that if the intersection of B with every 2-dimensional plane passing

through the origin is an ellipse, then B is an ellipsoid.

The `p spaces. If p ≥ 1 is a real number, the function

‖x‖p :=(

|x1|p + · · · + |xn|p)1/p

is a norm on Rn. When p tends to infinity, ‖x‖p converges to

‖x‖∞ := max|x1|, . . . , |xn|.

The normed space (Rn, ‖ · ‖p), 1 ≤ p ≤ ∞, is denoted by `np .

The unit ball of `n∞ is the n-dimensional cube of side length two, while the

unit ball of `n1 is the n-dimensional cross-polytope. In general, norms whose unit

balls are polytopes are called crystalline norms.

Subspaces of L1([0, 1], dx). The space of measurable functions f : [0, 1] → R

satisfying

‖f‖ :=

∫ 1

0

|f(x)| dx < ∞

is a normed space denoted by L1([0, 1], dx). The geometry of finite-dimensional

subspaces of L1([0, 1], dx) is closely related to problems of volume, area, and

integral geometry on normed and Finsler spaces. In the next few paragraphs,


we will summarize the properties of these subspaces that will be used in the

rest of the paper. For proofs, references, and to learn more about hypermetric

spaces, we recommend the landmark paper [Bolker 1969], as well as the surveys

[Schneider and Weil 1983] and [Goodey and Weil 1993].

First we begin with a beautiful metric characterization of the subspaces of

L1([0, 1], dx).

Definition 2.3. A metric space (M,d) is said to be hypermetric if it satisfies the

following stronger version of the triangle inequality: If m1, . . . ,mk are elements

of M and b1, . . . , bk are integers with∑

i bi = 1, then

k∑

i,j=1

d(mi,mj)bibj ≤ 0.

Theorem 2.4. A finite-dimensional normed space is hypermetric if and only if

it is isometric to a subspace of L1([0, 1], dx).

An important analytic characterization of a hypermetric normed space can be

given through the Fourier transform of its norm:

Theorem 2.5. A norm on Rn is hypermetric if and only if its distributional

Fourier transform is a nonnegative measure.

The characterizations above, important as they are, are hard to grasp at first

sight. A much more visual approach will be given after we review the duality of

normed spaces.

Minkowski spaces. Minkowski spaces are normed spaces with strict smoothness

and convexity properties. In precise terms, a norm ‖ · ‖ on a vector space X is

said to be a Minkowski norm if it is smooth outside the origin and the Hessian

of the function ‖ · ‖2at every nonzero point is a positive-definite quadratic form.

The unit sphere of a Minkowski space X is a smooth convex hypersurface SX

such that for any Euclidean structure on X the principal curvatures of SX are

positive.

2.1. Maps between normed spaces. An important feature of the geometry

of normed spaces is that the space of linear maps between two normed spaces

carries a natural norm.

Definition 2.6. If T : X → Y is a linear map between normed spaces

(X, ‖ · ‖X) and (Y, ‖ · ‖Y ), we define the norm of T as the supremum of ‖Tx‖Y

taken over all vectors x ∈ X with ‖x‖X ≤ 1.

A linear map T : X → Y is said to be short if its norm is less than or equal to one.

In other words, a short linear map does not increase distances. Two important

types of short linear maps between normed spaces are isometric embeddings and

isometric submersions:


Definition 2.7. An injective linear map T : X → Y between normed spaces

(X, ‖ · ‖X) and (Y, ‖ · ‖Y ) is said to be an isometric embedding if ‖Tx‖Y = ‖x‖X

for all vectors x ∈ X.

Definition 2.8. A surjective linear map T : X → Y between normed spaces

(X, ‖ · ‖X) and (Y, ‖ · ‖Y ) is said to be an isometric submersion if

‖Tx‖Y = inf ‖v‖X : v ∈ X and Tv = Tx


In terms of the unit balls, T : X → Y is an isometric embedding if and only if

T (BX) = T (X) ∩BY , and T is an isometric submersion if and only if T (BX) =

BY .

2.2. Dual spaces and polar bodies. From the previous paragraph, we see

that if (X, ‖ · ‖) is a normed space, then the set of all linear maps onto the one-

dimensional normed space (R, | · |) carries a natural norm. The resulting normed

space is called the dual of (X, ‖ · ‖) and is denoted by (X∗, ‖ · ‖∗). It is easy

to see that the double dual (i.e., the dual of the dual) of a finite-dimensional

normed space can be naturally identified with the space itself. The unit ball of

(X∗, ‖ · ‖∗) is said to be the polar of the unit ball of (X, ‖ · ‖).

Example. Holder’s inequality implies that if p > 1, the dual of `np is `n

q , where

1/p + 1/q = 1. Likewise, it is easy to see that the dual of `n1 is `n

∞.

If T : X 7→ Y is a linear map then the dual map T ∗ : Y ∗ 7→ X∗ is defined by

(T ∗ξ)(x) = ξ(Tx).

Note that ‖T ∗‖ = ‖T‖.

Exercise 2.9. Show that if T : X → Y is an isometric embedding between

normed spaces X and Y , the dual map T ∗ : Y ∗ → X∗ is an isometric submersion.

Many of the geometric constructions in convex geometry and the geometry of

normed spaces are functorial. More precisely, if we denote by N the category

whose objects are finite-dimensional normed spaces and whose morphisms are

short linear maps, many classical constructions define functors from N to itself.

Exercise 2.10. Show that the assignment (X, ‖ · ‖) 7→ (X∗, ‖ · ‖∗) is a con-

travariant functor from N to N.

Duals of hypermetric normed spaces. As advertised earlier in this section, the no-

tion of duality allows us to give a more geometric characterization of hypermetric

spaces.

Definition 2.11. A polytope in a vector space X is said to be a zonotope if all

of its faces are symmetric. A convex body is said to be a zonoid if it is the limit

(in the Hausdorff topology) of zonotopes.


Notice that an n-dimensional cube, as well as all its linear projections, are zono-

topes. In fact, it can be shown that any zonotope is the linear projection of a

cube (see, for example, Theorem 3.3 in [Bolker 1969]).

Theorem 2.12. Let X be a finite-dimensional normed space with unit ball BX .

The dual of X is hypermetric if and only if BX is a zonoid .

Notice that this immediately implies that the space `n1 , n ≥ 1, is hypermetric.

Duality in Minkowski spaces. If (X, ‖ · ‖) is a Minkowski space, the differential

of the function L := ‖ · ‖2/2,

dL(x)(y) :=1

2

d

dt‖x + ty‖2

t=0,

is a continuous linear map from X to X∗ that is smooth outside the origin and

homogeneous of degree one. This map is usually called the Legendre transform,

although that term is also used to describe some related concepts (see, for ex-

ample, § 2.2 in [Hormander 1994]). The following exercise describes the most

important properties of the Legendre transform.

Exercise 2.13. Let (X, ‖ · ‖) be a Minkowski space and let

L : X \ 0 → X∗ \ 0

be its Legendre transform.

(1) Show that if x ∈ X is a unit vector, then L(x) is the unique covector ξ ∈ X ∗

such that the equation ξ ·y = 1 describes the tangent plane to the unit sphere

SX at the point x.

(2) Show that the Legendre transform defines a diffeomorphism between the

unit sphere and its polar.

(3) Show that the inverse of the Legendre transform from X \0 to X∗ \0 is just

the Legendre transform from X∗ \ 0 to X \ 0.

(4) Show that the Legendre transform is linear if and only if X is a Euclidean

space.

Exercise 2.14. Show that a normed space is a Minkowski space if its unit

sphere and the unit sphere of its dual are smooth.

2.3. Sociology of normed spaces. If ‖ · ‖1 and ‖ · ‖2 are two norms on a

finite-dimensional vector space X, it is easy to see that there are positive numbers

m and M such that

m‖ · ‖2 ≤ ‖ · ‖1 ≤ M‖ · ‖2.

If we take the numbers m and M such that the inequalities are sharp, then

log(M/m) is a good measure of how far away one norm is from the other.

For example, the following well-known result states that we can always ap-

proximate a norm by one whose unit sphere is a polytope or by one such that

its unit sphere and the unit sphere of its dual are smooth.


Proposition 2.15. Let ‖ · ‖ be a norm on the finite-dimensional vector space

X. Given ε > 0, there exist a crystalline norm ‖ · ‖1 and a Minkowski norm

‖ · ‖2 on X such that

‖ · ‖1 ≤ ‖ · ‖ ≤ (1 + ε)‖ · ‖1,

‖ · ‖2 ≤ ‖ · ‖ ≤ (1 + ε)‖ · ‖2.

For a short proof see Lemma 2.3.2 in [Hormander 1994] .

In many circumstances, one wants to measure how far is one normed space

from being isometric to another. The straightforward adaptation of the previous

idea leads us to the following notion:

Definition 2.16. The Banach–Mazur distance between n-dimensional normed

spaces X and Y , is the infimum of the numbers log(‖T‖‖T−1‖), where T ranges

over all invertible linear maps from X to Y .

Notice that the Banach–Mazur distance is a distance on the set of isometry

classes of n-dimensional normed spaces: two such spaces are at distance zero if

and only if they are isometric.

An important question is to determine how far a general n-dimensional normed

space is from being Euclidean. The answer rests on two results of independent

interest:

Theorem 2.17 (Loewner). If B is a convex body in an n-dimensional vector

space X, there exists a unique n-dimensional ellipsoid E ⊂ B such that for any

Lebesgue measure on X, the ratio vol(B)/vol(E) is minimal .

Theorem 2.18 [John 1948]. Let X be an n-dimensional normed space with unit

ball B. If E ⊂ B is the Loewner ellipsoid of B, then

E ⊂ B ⊂√

nE.

Exercise 2.19. Show that the Banach–Mazur distance from an n-dimensional

normed space to a Euclidean space is at most log(n)/2.

The structure of the set of isometry classes of n-dimensional normed spaces is

given by the following theorem (see [Thompson 1996, page 73] for references and

some of the history on the subject):

Theorem 2.20. The set of isometry classes of n-dimensional normed spaces,

Mn, provided with the Banach–Mazur distance is a compact , connected metric

space.

The Banach–Mazur compactum, Mn, enters naturally into Finsler geometry by

the following construction: Let π : ζ → M be a vector bundle with n-dimensional

fibers such that every fiber ζm = π−1(m) carries a norm that varies continuously

with the base point (a Finsler bundle). The (continuous) map

I : M −→ Mn


that assigns to each point m ∈ M the isometry class of ζm measures how the

norms vary from point to point.

Currently, there are not many results that describe the map I under different

geometric and/or topological hypotheses on the bundle. However the following

exercise (and its extension in [Gromov 1967]) shows that such results are possible.

Exercise 2.21. Let π : ζ → S2 be a Finsler bundle whose fibers are 2-

dimensional. Show that if the bundle is nontrivial and the map I is constant,

then the image of S2 under I is the isometry class of 2-dimensional Euclidean

spaces.

A corollary of this exercise is that if X is a three-dimensional normed space

such that all its two-dimensional subspaces are isometric, then X is Euclidean.

Another interesting corollary is that a Berwald (Finsler) metric on S2 must be

Riemannian.

3. Volumes on Normed Spaces

In defining the notion of volume on normed spaces, it is best to adopt an

axiomatic approach. We shall impose some minimal set of conditions that are

reasonable and then try to find out to what extent they can be satisfied, and to

what point they determine our choices.

In a normed space, all translations are isometries. This suggests that we

require the volume of a set to be invariant under translations. Since any finite-

dimensional normed space is a locally compact, commutative group, we can apply

the following theorem of Haar:

Theorem 3.1. If µ is a translation-invariant measure on Rn for which all

compact sets have finite measure and all open sets have positive measure, then

µ is a constant multiple of the Lebesgue measure.

Proofs of this theorem can be found in many places. A full account is given in

[Cohn 1980] and an abbreviated version in [Thompson 1996].

In the light of Haar’s theorem, in order to give a definition of volume in

every normed space, we must assign to every normed space X a multiple of the

Lebesgue measure. Since the Lebesgue measure is not intrinsically defined (it

depends on a choice of basis for X), it is best to describe this assignment as

a choice of a norm µ in the 1-dimensional vector space ΛnX, where n is the

dimension of X; if x1, . . . ,xn ∈ X, we define µ(x1 ∧x2 ∧ · · ·∧xn) as the volume

of the parallelotope formed by these vectors.

Another natural requirement is monotonicity: if X and Y are n-dimensional

normed spaces and T : X → Y is a short linear map (i.e., a linear map that does

not increase distances), we require that T does not increase volumes. Notice

that this implies that isometries between normed spaces are volume-preserving.


The monotonicity requirement makes a definition of volume on normed spaces

into a functor from N to itself that takes the n-dimensional normed space

(X, ‖ · ‖) to the 1-dimensional normed space (ΛnX,µ). While we shall often

abandon this viewpoint, it is a guiding principle throughout the paper with

which we would like to acquaint the reader early on.

Definition 3.2. A definition of volume on normed spaces assigns to every

n-dimensional, n ≥ 1, normed space X a normed space (ΛnX,µX) with the

following properties:

(1) If X and Y are n-dimensional normed spaces and T : X → Y is a short

linear map , then the induced linear map T∗ : ΛnX → ΛnY is also short.

(2) The map X 7→ (ΛnX,µX) is continuous with respect to the topology induced

by the Banach–Mazur distance.

(3) If X is Euclidean, then µX is the standard Euclidean volume on X.

Before presenting the principal definitions of volume in normed spaces, let us

make the first link between these concepts and the affine geometry of convex

bodies.

Exercise 3.3. Assume we have a definition of volume in normed spaces and

use it to assign a number to any centrally symmetric convex body B ⊂ Rn by

the following procedure: Consider Rn as the normed space X whose unit ball is

B and compute

V(B) := µX(B) =

∫

B

µX .

Show that if T : Rn → R

n is an invertible linear map, then V(B) = V(T (B)),

and write the monotonicity condition in terms of the affine invariant V.

Notice that we can turn the tables and start by considering a suitable affine

invariant V of centered convex bodies and give a definition of volume in normed

spaces by prescribing that the volume of the unit ball B of a normed space X

be given by V(B).

Exercise 3.4. Let µ be a definition of volume for 2-dimensional normed spaces.

Use John’s theorem to show that if B is the unit disc of a two-dimensional normed

space X, then π/2 ≤ µX(B) ≤ 2π.

3.1. Examples of definitions of volume in normed spaces. The study

of the four definitions of volume we shall describe below makes up the most

important part of the theory of volumes on normed and Finsler spaces.

The Busemann definition. The Busemann volume of an n-dimensional normed

space is that multiple of the Lebesgue measure for which the volume of the unit

ball equals the volume of the Euclidean unit ball in dimension n, εn, . In other

words, we have chosen as our affine invariant the constant εn, where n is the

dimension of the space.


Another way to define the Busemann volume of a normed space X is by setting

µb(x1 ∧ x2 ∧ · · · ∧ xn) =εn

vol(B; x1 ∧ x2 ∧ · · · ∧ xn),

where the notation vol(B; x1 ∧ x2 ∧ · · · ∧ xn) indicates the volume of B in the

Lebesgue measure determined by the basis x1, . . . ,xn.

Using Brunn–Minkowski theory, Busemann showed in [1947] that the Buse-

mann volume of an n-dimensional normed space equals its n-dimensional Haus-

dorff measure. Hence, from the viewpoint of metric geometry, this is a very

natural definition.

Exercise 3.5. Show that the Busemann definition of volume satisfies the axioms

in Definition 3.2.

The Holmes–Thompson definition. Let X be an n-dimensional normed space and

let B∗ ⊂ X∗ be the dual unit ball. If x1, . . . ,xn is a basis of X and ξ1, . . . , ξn

is its dual basis, define

µht(x1 ∧ x2 ∧ · · · ∧ xn) := ε−1n vol(B∗; ξ1 ∧ ξ2 ∧ · · · ∧ ξn).

Another way of defining the Holmes–Thompson volume is by considering the

set B×B∗ in the product space X ×X∗. Since X×X∗ has a natural symplectic

structure defined by

ω((x1, ξ1), (x2, ξ2)) := ξ2(x1) − ξ1(x2),

it has a canonical volume (the symplectic or Liouville volume) defined by the

n-th exterior power ωn of ω, divided by n!. The Holmes–Thompson volume of

the n-dimensional normed space X is the multiple of the Lebesgue measure for

which the volume of the unit ball equals the Liouville volume of B ×B∗ divided

by the volume of the Euclidean unit ball of dimension n. We mention in passing

that in convex geometry it is usual to denote the Liouville volume of B ×B∗ as

the volume product of B, vp(B).

The Holmes–Thompson definition — introduced in [Holmes and Thompson

1979] — was originally motivated by purely geometric considerations. However,

from the physical point of view it is the natural definition of volume if we think

of normed spaces as homogeneous, anisotropic media: media in which the speed

of light varies with the direction of the light ray, but not with the point at which

the propagation of light originates.

It is interesting to remark that the Busemann definition and the Holmes–

Thompson definition are dual functors: to obtain the Holmes–Thompson volume

of an n-dimensional normed space X we pass to the dual normed space X∗, we

apply the “Busemann functor” to obtain (ΛnX∗, µbX∗) and then pass to the dual

of the normed space (ΛnX∗, µbX∗).

Exercise 3.6. Consider a definition of volume (X, ‖ · ‖) 7→ (ΛnX,µX), where

n is the dimension of X, and define its dual definition by the map (X, ‖ · ‖) 7→


(ΛnX,µ∗X) := (ΛnX∗, µX∗)∗. Show that µ∗ also satisfies the axioms in Defini-

tion 3.2.

The notion of duality is somewhat mysterious and is closely related to the duality

between intersections and projections proposed in [Lutwak 1988], and which led

to the development of the dual Brunn–Minkowski theory. We shall have a little

more to say about this duality after presenting a second dual pair of volume

definitions.

Gromov’s mass. If X is an n-dimensional normed space, define µm : ΛnX →[0,∞) by the formula

µm(a) := inf

n∏

i=1

‖xi‖ : x1 ∧ x2 ∧ · · · ∧ xn = a

.

Another way to define the mass of an n-dimensional normed space X is as the

multiple of the Lebesgue measure for which the volume of the maximal cross-

polytope inscribed to the unit ball is 2n/n!.

Exercise 3.7. Consider the 2-dimensional normed space whose unit disc D is

a regular hexagon. What is µm(D)?

The Benson definition or Gromov’s mass∗. One way to make the Benson defi-

nition is as the dual of mass: given an n-dimensional normed space X together

with a basis x1, . . . ,xn, we take the dual basis ξ1, . . . , ξn in X∗ and define

µm∗X (x1 ∧ x2 ∧ · · · ∧ xn) :=

1

µmX∗(ξ1 ∧ ξ2 ∧ · · · ∧ ξn)

.

This is Gromov’s definition [1983]. Benson [1962] originally defined the mass∗of an n-dimensional normed space as the multiple of the Lebesgue measure for

which the volume of a minimal parallelotope circumscribed to the unit ball is

2n.

Exercise 3.8. Consider the 2-dimensional normed space whose unit disc D is

a regular hexagon. What is µm∗(D)?

The following exercise gives a third characterization of mass∗.

Exercise 3.9. Let X be an n-dimensional normed space and let B be its unit

ball. A basis ξ1, . . . , ξn of X∗ is said to be short if |ξi(x)| ≤ 1 for all x ∈ B

and all i, 1 ≤ i ≤ n (i.e., if all the vectors in the basis are in the dual unit ball).

Show that for any n-vector a ∈ ΛnX

µm∗(a) = sup|ξ1 ∧ ξ2 ∧ · · · ∧ ξn (a)| : ξ1, . . . , ξn is a short basis of X∗

It is not hard to come up with other definitions of volume. For example, instead

of considering inscribed cross-polytopes and circumscribed parallelotopes one

might consider maximal inscribed or minimal circumscribed ellipsoids (as in

Loewner’s theorem cited above) and then specify the volume of either to be


εn. However, as we shall see in the next two sections, a good definition of

volume must satisfy some additional conditions that are very hard to verify.

The examples given above are important mainly because their study provides a

common context to many problems in convex, integral, and differential geometry.

3.2. The volume of the unit ball. If we are given a definition of volume

and a normed space, we would like to compute the volume of the unit ball.

This is, of course, trivial if we work with the Busemann definition, but for the

other definitions it is a challenging problem. Let us start with some simple

experiments.

Example 3.10. In the table below we use 7 different norms in R3 whose unit

balls are, in order, the Euclidean unit ball; the cube with vertices at (±1,±1,±1);

the octahedron with vertices at ±(1, 0, 0), ±(0, 1, 0), ±(0, 0, 1); the right cylinder

over the unit circle in the xy-plane and with −1 ≤ z ≤ 1; its dual, the double

cone which is the convex hull of the unit circle in the xy-plane and the points

±(0, 0, 1); the affine image of the cuboctahedron that has vertices at ±(1, 0, 0),

±(0, 1, 0), ±(0, 0, 1), ±(1,−1, 0), ±(1, 0,−1), ±(0, 1,−1); and its dual, the affine

image of the rhombic dodecahedron, that has vertices at ±(1, 1, 1), ±(0, 1, 1),

±(1, 0, 1), ±(1, 1, 0), ±(0, 0, 1), ±(0, 1, 0), ±(1, 0, 0). These are listed in the first

column. In the subsequent columns are the volumes of each unit ball using the

different definitions of volume.

The ball B µb(B) µht(B) µm∗(B) µm(B)

ball 4π/3 4π/3 4π/3 4π/3

cube 4π/3 8/π 8 2

octahedron 4π/3 8/π 16/3 4/3

cylinder 4π/3 π 2π π

double cone 4π/3 π 4π/3 2π/3

cuboctahedron 4π/3 10/π 20/3 10/3

rhombic dodecahedron 4π/3 10/π 4 2

Exercise 3.11. Verify these numbers.

Given a definition of volume, an interesting problem is to determine sharp up-

per and lower bounds for the volume of the unit balls of n-dimensional normed

spaces. In the case of the Holmes–Thompson definition, this question has a clas-

sical reformulation: give sharp upper and lower bounds for the volume product

of an n-dimensional centrally-symmetric convex body.

Theorem 3.12 (Blaschke–Santalo inequality). The Holmes–Thompson

volume of the unit ball of an n-dimensional normed space is less than or equal to

the volume of the Euclidean unit ball of dimension n. Moreover , equality holds

if and only if the space is Euclidean.

The sharp lower bound for the Holmes–Thompson volume of unit balls is a

reformulation of a long-standing conjecture of Mahler [1939]:


Conjecture. The Holmes–Thompson volume of the unit ball of an n-dimen-

sional normed space is greater than or equal to 4n/εnn!. Moreover , equality holds

if and only if the unit ball is a parallelotope or a cross-polytope.

This conjecture has been verified by Mahler [1939] in the two-dimensional case

and by Reisner [1985, 1986] in the case when either the normed space or its dual

is hypermetric.

For µm∗(B) the upper bound of 2n is attained for a parallelotope and for

µm(B) the equivalent lower bound of 2n/n! is attained by a cross-polytope. One

also has µm∗(B) ≥ 2n/n! and µm(B) ≤ 2n but these are far from optimal; better

bounds will be obtained after studying the relationship between the different

definitions of volume.

3.3. Relationship between the definitions of volume. There are several

relationships between the various measures we are considering. For example, the

Blaschke–Santalo inequality is clearly equivalent to the following theorem:

Theorem 3.13. If X is an n-dimensional normed space, then µhtX ≤ µb

X with

equality if and only if X is Euclidean.

For mass and mass∗ we have the following inequality:

Proposition 3.14. If X is an n-dimensional normed space, then µmX ≤ µm∗

X .

Proof. Let P be a minimal circumscribed parallelotope to the unit ball B.

Then (see for example [Thompson 1999], but there are many other possible

references) the midpoint of each facet of P is a point of contact with B. The

convex hull of these midpoints is a cross-polytope C inscribed to B. Also, if

P is given the volume 2n, then C has volume 2n/n!. Hence, in this situation,

a maximal inscribed cross-polytope will have volume greater than or equal to

2n/n!.

Theorem 3.15. If X is an n-dimensional normed space, then µmX ≤ µb

X with


The proof depends on the following theorem of McKinney [1974]:

Theorem 3.16. Let K ⊂ X be a convex set symmetric about the origin and let

S be a maximal simplex contained in K with one vertex at the origin, then for

any Lebesgue measure λ

λ(S)/λ(K) ≥ 1/n!εn

with equality if and only if K is an ellipsoid .

Proof of Theorem 3.15. If B is the unit ball of X then µb(B) = εn and

µm(B) = 2nλ(B)/n!λ(C) where C is a maximal cross-polytope inscribed to

B. Moreover, C is the convex hull of S ∪ −S, where S is a maximal simplex

inscribed to B with one vertex at the origin. It follows from the theorem that

λ(C)/λ(B) ≥ 2n/n!εn which, upon rearrangement, gives µb(B) ≥ µm(B).


The relationship between mass∗ and the Holmes–Thompson volume follows from

Theorem 3.15 and the following simple exercise:

Exercise 3.17. Let µ and ν be two definitions of volume, and let µ∗ and ν∗ be

their dual definitions. Show that if for every normed space X

µX ≤ νX , then ν∗X ≤ µ∗

X .

Corollary 3.18. If X is an n-dimensional normed space, then µhtX ≤ µm∗

X with


The previous inequalities are summarized by the diagram

µb

µht

µm∗

µm? ?

@@

@@

@@R

Notice that as a consequence of the Mahler–Reisner inequality we have the fol-

lowing lower bounds for the mass and mass∗ of unit balls in normed spaces and

their duals.

Corollary 3.19. For any unit ball B, we have µm(B) ≤ εn and , if B is either

a zonoid or the dual of a zonoid , µm∗(B) ≥ 4n/n!εn.

Problem. Is the mass∗ of the unit ball of an n-dimensional normed space at

least 4n/n!εn? This is a weaker version of Mahler’s conjecture.

3.4. Extension to Finsler manifolds

Definition 3.20. A volume density on ann-dimensional manifold M is a con-

tinuous function

Φ : ΛnTM −→ R

such that for every point m ∈ M the restriction of Φ to ΛnTmM is a norm. A

volume density is said to be smooth if the function Φ is smooth outside the zero

section.

If M is an oriented manifold and Φ is a volume density on M , then we can define

a volume form Ω on M whose value at a basis x1, . . . ,xn of TmM is Φ(m; x1 ∧x2 ∧ · · · ∧xn) if the basis is positively oriented and −Φ(m; x1 ∧x2 ∧ · · · ∧xn) if

it is negatively oriented. For any positively oriented n-dimensional submanifold

U ⊂ M we have that∫

U

Φ =

∫

U

Ω.

However, the integral of a volume density does not depend on the orientation and

volume densities can be defined in nonorientable manifolds like the projective

plane where no volume form exists.


Definition 3.21. A continuous Finsler metric F on a manifold M assigns a

norm, F (m, ·), to each tangent space TmM in such a way that the norm varies

continuously with the base point. A continuous Finsler manifold is a pair (M,F ),

where M is a manifold and F is a continuous Finsler metric on M .

An important class of examples of Finsler manifolds are finite-dimensional sub-

manifolds of normed spaces. If M is a submanifold of a finite-dimensional normed

space X, at each point m ∈ M the tangent space TmM can be thought of as a

subspace of X and, as such, it inherits a norm.

If γ : [a, b] → M is a differentiable curve on a continuous Finsler manifold

(M,F ), we define

length(γ) :=

∫ b

a

F (γ(t), γ(t)) dt.

This definition can be extended in the obvious way to piecewise-differentiable

curves. If x and y are two points in M , we define the distance between x and y

as the infimum of the lengths of all piecewise-differentiable curves that join them.

Thus, continuous Finsler manifolds are metric spaces and metric techniques can

be used to study them.

Each definition of volume on normed spaces gives a definition of volume on

continuous Finsler manifolds: if we are given a volume definition µ and an n-

dimensional continuous Finsler manifold (M,F ), then the map that assigns to

every point m the norm µTmM on ΛnTmM is a volume density on M . Notice

that, in particular, a definition of volume on normed spaces immediately yields

a way to measure the volumes of submanifolds of a normed space X because, as

remarked above, the tangent space TmM of such a submanifold can be regarded

as a subspace of X and so inherits both a norm and a volume. This will be

studied from an extrinsic viewpoint and in much more detail in the next section.

Exercise 3.22. Show that if a definition of volume on normed spaces is used

to define a volume on Finsler manifolds, it satisfies the following two properties:

(1) If the Finsler manifold is Riemannian, its volume is the standard Riemannian

volume;

(2) If ϕ : M → N is a short map (i.e., does not increase distances) between two

Finsler manifolds of the same dimension, then ϕ does not increase volumes.

Extending our four volume definitions from normed spaces to continuous Finsler

manifolds, we may speak of the Busemann, Holmes–Thompson, mass, and mass∗definition of volumes on continuous Finsler manifolds. To end the section we re-

late the Busemann and Holmes–Thompson definitions with well-known geometric

constructions.

As was previously remarked, Finsler manifolds are metric spaces and, as

such, we can define their volume as their Hausdorff measure: if (M,F ) is an

n-dimensional Finsler manifold and r > 0, we cover M by a family of metric

balls of radius at most r, B(m1, r1), B(m2, r2), . . ., and consider the quantity


εn(rn1 + rn

2 + · · ·). We now take the infimum of this quantity over all possible

covering families and take the limit as r tends to zero. The resulting number is

the n-dimensional Hausdorff measure of (M,F ).

Theorem 3.23 [Busemann 1947]. The Busemann volume of a continuous Finsler

manifold is equal to its Hausdorff measure.

To explain the second construction, we need to recall some standard facts about

the geometry of cotangent bundles.

If π : T ∗M → M is the canonical projection, we define the canonical 1-form

α on T ∗M by the formula

α(vp) := p(π∗(vp)).

In standard coordinates (x1, . . . , xn, p1, . . . , pn), α :=∑n

i=1 pidqi. The canonical

symplectic form on T ∗M is defined as ω := −dα and the Liouville volume is

defined by Ω := ωn/n!.

If (M,F ) is a continuous Finsler manifold, each tangent space TmM carries

the norm F (m, ·) and, hence, each cotangent space T ∗mM carries the dual norm

F ∗(m, ·). Let us denote the unit ball in T ∗mM by B∗

m and define the unit co-disc

bundle of M as the set

B∗(M) :=⋃

m∈M

B∗m ⊂ T ∗M.

Proposition 3.24. The Holmes–Thompson volume of an n-dimensional , con-

tinuous Finsler manifold is equal to the Liouville volume of its unit co-disc bundle

divided by the volume of the n-dimensional Euclidean unit ball .

Proof. It suffices to verify the result on normed spaces where it easily follows

from the definitions.

Theorem 3.25 [Duran 1998]. If M is a Finsler manifold , then the Holmes–

Thompson volume of M is less than or equal to its Hausdorff measure with

equality if and only if M is Riemannian.

Proof. By the Blaschke–Santalo inequality, at each point m ∈ M we have that

µhtTmM ≤ µb

TmM with equality if and only if TmM is Euclidean. The result now

follows immediately from Theorems 3.23 and 3.24 .

4. k-Volume Densities

The theory of volumes and areas on Euclidean and Riemannian spaces is

based on the fact that a Euclidean structure on a vector space induces natural

Euclidean structures on its exterior powers: if x1, . . . ,xn is an orthonormal basis

of a Euclidean space X, then the vectors

xi1 ∧ xi2 ∧ · · · ∧ xik, 1 ≤ i1 < i2 < · · · < ik ≤ n,


form an orthonormal basis of ΛkX. If we want to know the area of the parallel-

ogram formed by the vectors x and y, we need to compute the norm of x∧ y in

Λ2X. In normed and Finsler spaces these simple algebraic constructions, which

should be seen as functors from the category of Euclidean spaces onto itself, can-

not be reproduced and we need to understand their geometry to see how they

may be naturally extended to these spaces.

The first important remark is that in order to compute k-dimensional volumes

(k-volumes from now on), we do not need to define a norm on all of ΛkX. It

suffices to define the magnitude of vectors of the form v1 ∧ v2 ∧ · · · ∧ vk, where

v1, . . . ,vk are vectors in X. In this paper we shall refer to these k-vectors as

simple and denote the set of all simple k-vectors in X by ΛksX. Note that for

k = 1, n− 1 every k-vector is simple, which makes the study of (n−1)-volume in

n-dimensional normed spaces a richer and more approachable subject than the

study of volumes in higher codimension. Indeed, when k 6= 1, n − 1, ΛksX is not

a vector subspace of ΛkX, but just an algebraic cone.

Exercise 4.1. Let Q : Λ2R

4 → Λ4R

4 be the quadratic form defined by Q(a) =

a ∧ a. Show that Λ2sR

4 is the quadric Q = 0 and use this to prove that the

intersection of Λ2sR

4 with the (Euclidean) unit sphere in Λ2R

4 is a product of

two 2-dimensional spheres.

In general, the intersection of ΛksR

n with the Euclidean unit sphere in ΛkR

n is

the Plucker embedding of the Grassmannian of oriented k-planes in Rn, G+

k (Rn).

Let us recall that this Grassmannian is a smooth manifold of dimension k(n−k).

Following our first remark, we see that in order to compute k-volumes in a

normed space X, we need to define a “norm” on the cone of simple k-vectors

of X. The fact that ΛksX is not a vector space complicates matters since it is

not clear how to write the triangle inequality, and, even if an apparent analogue

could be found, it would have to be justified in terms of its geometric significance.

Nevertheless, the homogeneity and positivity of a norm are easy to generalize:

Definition 4.2. A k-density on a vector space X is a continuous function

φ : ΛksX −→ R

that is homogeneous of degree one (i.e., φ(λa) = |λ|φ(a)). A k-density φ is said

to be a k-volume density if φ(a) ≥ 0 with equality if and only if a = 0.

4.1. Examples of k-volume densities. In the previous chapter we studied

four natural volume definitions on normed spaces. Each one of these definitions

yields natural constructions of k-volume densities on the spaces of simple k-

vectors of a normed space X.

Given a volume definition µ and a k-vector a in a k-dimensional normed space

Y , we can compute µ(a). To be perfectly rigorous, we should include the normed

space as a variable in µ, for example, by writing µY (a). If a is a simple k-vector

in an n-dimensional normed space X, then we may consider it as a k-vector on


the k-dimensional normed space “spanned by a”,

〈a〉 := x ∈ X : a ∧ x = 0 ⊂ X,

(provided with the induced norm), and compute µ〈a〉(a). Thus, once we have

chosen a way to define volume in all finite-dimensional normed spaces, we have

a way to associate to each norm on an n-dimensional vector space X a family of

k-volume densities, with 1 ≤ k ≤ n.

The Busemann k-volume densities. Let X be a normed space of dimension n

with unit ball B, and let k be a positive integer less than n. The Busemann

k-volume density on X is defined by the formula

µb(a) :=εk

vol(B ∩ 〈a〉; a).

The Holmes–Thompson k-volume densities. Let X be a normed space of dimen-

sion n, and let k be a positive integer less than n. If a is a simple k-vector

spanning the k-dimensional subspace 〈a〉, we consider the inclusion of 〈a〉 into

X and the dual projection π : X∗ → 〈a〉∗. Regarding a as a volume form on

〈a〉∗ we define

µht(a) := ε−1k

∫

π(B∗

X)

|a|.

The mass k-volume densities. Let (X, ‖ · ‖) be a normed space of dimension n,

and let k be a positive integer less than n. The mass k-volume density on X is

defined by the formula

µm(a) := inf

k∏

i=1

‖xi‖ : x1 ∧ x2 ∧ · · · ∧ xk = a

.

The mass∗ k-volume densities. According to the characterization of mass∗ given

in Exercise 3.9, we may describe the mass∗ k-volume densities as follows:

Let X be a normed space and let W ⊂ X be a k-dimensional subspace. If a

is a simple k-vector on X, we define µm∗(a) as the supremum of the numbers

|ξ1 ∧ ξ2 ∧ · · · ∧ ξk(a)|, where ξ1, . . . , ξk ranges over all short bases of 〈a〉∗.

However, there is a simpler description:

Exercise 4.3. Using the Hahn–Banach theorem and the notation above, show

that µm∗(a) is the supremum of the numbers |ξ1 ∧ ξ2 ∧ · · · ∧ ξk(a)|, where

ξ1, . . . , ξk ∈ B∗X .

In the study of volumes and areas on Finsler manifolds, we shall also need to

work with k-densities and smooth k-densities on manifolds. For this purpose we

introduce the bundle of simple tangent k-vectors on a manifold M , ΛksTM . This

is a subbundle of algebraic cones of the vector bundle ΛkTM , and if we omit the

zero section it is a smooth manifold.


Definition 4.4. A k-density φ (resp. k-volume density) on a manifold M is a

continuous function φ : ΛksTM → R such that at each point m, the restriction of

φ to ΛksTmM is a k-density (resp. k-volume density). If the function φ is smooth

outside the zero section, we shall say that the density is smooth.

Since every tangent space of a Finsler manifold (M,F ) is a normed space, we may

define the Busemann, Holmes–Thompson, mass, and mass∗ k-volume densities

on (M,F ) by assigning to each tangent space its respective k-density. It is easy

to show that if F is a smooth Finsler metric, then the Busemann and Holmes–

Thompson k-volume densities are smooth. This is probably not the case with

mass and mass∗, but we have no explicit examples to illustrate this.

Just like differential k-forms, k-densities can be pulled back: if f : N → M is

a smooth map and φ is a k-density on M , then

f∗φ(v1 ∧ v2 ∧ · · · ∧ vk) = φ(

Df(v1) ∧ Df(v2) ∧ · · · ∧ Df(vk))

.

Remark that if f : N → M is an immersion and φ is a k-volume density on M ,

then f∗φ is a k-volume density on N .

Also like differential forms, k-densities can be integrated over k-dimensional

submanifolds: if N ⊂ M is a k-dimensional submanifold of M and i : N → M

is the inclusion map, then i∗φ is a volume density on N , and its integral over N

was defined in Section 3. This integral is independent of the parameterization

and orientation of N . In the same way, we may define the integral of a k-density

over a k-chain.

For the rest of the chapter, we associate to a given k-volume density φ on a

vector space X the functional

N 7−→∫

N

φ,

and investigate the relationship between the behavior of the functional and cer-

tain convexity properties of φ. The easiest case is when φ is an (n−1)-volume

density in an n-dimensional vector space.

4.2. Convexity of (n−1)-volume densities. This case is special because

every (n−1)-vector in an n-dimensional vector space, X, is simple and we may

impose the condition that an (n−1)-volume density be a norm in Λn−1X. This

is, for example, satisfied by (n−1)-volume densities for the Busemann, Holmes–

Thompson, and mass∗ definitions of volume. Nevertheless, it remains to see why

such a condition is desirable.

The next result is the first of four characterizations of norms on Λn−1X.

Theorem 4.5. Let φ be an (n−1)-volume density on an n-dimensional vector

space X. The following conditions on φ are equivalent :


• φ is a norm;

• If P ⊂ X is a closed (n−1)-dimensional polyhedron in X, then the area of any

one of its facets is less than or equal to the sum of the areas of the remaining

facets.

Before proving this theorem, we need to introduce a classical construction that

associates to any k-dimensional polyhedral surface on X a set of simple k-vectors.

This set will be called the Gaussian image of the polyhedron (see also [Burago

and Ivanov 2002], where the almost identical notion of Gaussian measure is

used).

If P ⊂ X is a polyhedron with facets F1, . . . , Fm we associate to each facet Fi

the simple k-vector ai such that 〈ai〉 is parallel to Fi and such that vol(Fi; ai) = 1.

The Gaussian image of P is the set a1, . . . , am ⊂ ΛksX. If φ is a k-volume

density in X, the k-volume of P (with respect to φ) is just φ(a1) + · · · + φ(am).

Exercise 4.6. Show that if a1, . . . , am ⊂ ΛksX is the Gaussian image of a

closed polyhedron in X, then a1 + · · · + am = 0.

In general, the condition that the sum of a set of simple k-vectors be zero, does

not imply that it is the Gaussian image of a closed k-dimensional polyhedron in

X. However, in codimension one we have the following celebrated theorem of

Minkowski.

Theorem 4.7 (Minkowski). A set of (n−1)-vectors of an n-dimensional vector

space X is the Gaussian image of a closed , convex polyhedron if and only if the

(n−1)-vectors span Λn−1X and their sum equals zero.

To prove Theorem 4.5, we shall need an easy particular case of Minkowski’s

result:

Exercise 4.8. Let X be an n-dimensional vector space and let a1, . . . , an be a

basis of Λn−1X. Show that there exists a simplex in X whose Gaussian image

is the set a1, . . . , an,−(a1 + · · · + an).

Proof of Theorem 4.5. Assume that φ is a norm, and let P ⊂ X be an (n−1)-

dimensional closed polyhedron with Gaussian image a0, a1, . . . , am. Since the

sum of the ai’s is zero, we may use the triangle inequality to write

φ(a0) = φ(a1 + · · · + am) ≤ φ(a1) + · · · + φ(am).

In other words, the area of the facet corresponding to a0 is less than or equal to

the sum of the areas of the remaining facets.

To prove the converse, we take any two (n−1)-vectors a1 and a2, which we

assume to be linearly independent, and use them as part of a basis a1, . . . , an of

Λn−1X. By Exercise 4.8, the set a1, . . . , an,−(a1 + · · · + an) is the Gaussian

image of a simplex in X. Then, by assumption,

φ(a1 + · · · + an) ≤ φ(a1) + · · · + φ(an).


By letting a3, . . . , an tend to zero in the above inequality we obtain the triangle

inequality φ(a1 + a2) ≤ φ(a1) + φ(a2), and, therefore, φ is a norm.

Exercise 4.9. Consider the tetrahedron in the normed space `3∞ with vertices

(0, 0, 0), (−1, 1, 1), (1,−1, 1), (1, 1,−1), and show that the mass of the facet

opposite the origin is greater than the sum of the masses of the three other

facets. Hint. Use the definition of the mass 2-volume density in terms of minimal

circumscribed parallelograms.

By Theorem 4.5, the previous exercise shows that the mass (n−1)-volume density

of a normed space X is not necessarily a norm in Λn−1X. As we shall see in

the rest of this chapter, this is a good reason to disqualify mass as a satisfactory

definition of volume on normed spaces.

Our second characterization of norms in Λn−1X is another variation on the

theme of flats minimize.



• φ is a norm;

• Whenever C and C ′ are (n−1)-chains with real coefficients such that ∂C =

∂C ′ and the image of C is contained in a hyperplane, then the area of C is

less than or equal to the area of C ′.

In order to prove this theorem we need to introduce the concept of calibration

formalized by Harvey and Lawson [1982].

Definition 4.11. A closed k-form ω is said to calibrate a k-density φ if for all

simple k-vectors a in TM we have that ω(a) ≤ φ(a) and equality is attained on

a nonempty subset of ΛksTM .

The homogeneity of ω and φ allows us to consider the set where they coincide as

a subset E of the bundle of oriented k-dimensional subspaces of TM , G+k (TM).

Proposition 4.12 [Harvey and Lawson 1982]. Let φ be a k-volume density on

a manifold M , let ω be a closed k-form on M that calibrates φ and let E ⊂G+

k (TM) be the set where φ and ω coincide. If N ⊂ M is a k-dimensional

oriented submanifold all of whose tangent planes belong to E, and N ′ is another

submanifold of M homologous to N , then

∫

N

φ ≤∫

N ′

φ.

Proof. Using that φ = ω on the tangent spaces of N and Stokes’ formula, we

have∫

N

φ =

∫

N

ω =

∫

N ′

ω ≤∫

N ′

φ.


Proof of Theorem 4.10. Assume that φ is a norm, let C and C ′ be as in the

statement of the theorem, and let a be an (n−1)-vector on X such that φ(a) = 1

and the subspace 〈a〉 is parallel to the hyperplane containing the image of C.

Since the unit sphere in (Λn−1X,φ) is a convex hypersurface, it has a supporting

hyperplane that touches it at a. This hyperplane can be given as the set ω = 1,

where ω is a constant-coefficient (n−1)-form on X. Since the unit sphere lies in

the half-space ω ≤ 1, we have ω ≤ φ and, thus, ω calibrates φ.

Using that ω = φ on C, that dω = 0, and that C + (−C ′) is a closed chain

homologous to zero, we have∫

C

φ =

∫

C

ω =

∫

C′

ω ≤∫

C′

φ.

To prove the converse, note that the second condition in the theorem im-

mediately implies that the (n−1)-volume of the facet of a closed polyhedron is

less than or equal to the sum of the (n−1)-volumes of the remaining facets. By

Theorem 4.5, this implies that φ is a norm.

In Euclidean geometry, the orthogonal projection onto a k-dimensional subspace

is area-decreasing. This can be generalized as follows:



• φ is a norm;

• For every (n−1)-dimensional subspace W ⊂ X there is a φ-decreasing linear

projection PW : X → W .

The proof of this theorem rests on a simple lemma in multi-linear algebra.

Lemma 4.14. Let X be an n-dimensional vector space and let W ⊂ X be a

k-dimensional subspace. If w1, . . . ,wk is a basis of W and ω ∈ ΛkX∗ is such

that ω(w1 ∧ w2 ∧ · · · ∧ wk) = 1, then the linear map

Px :=

k∑

i=1

(−1)iω(x ∧ w1 ∧ · · · ∧ wi ∧ · · · ∧ wk)wi

is a projector with range W . Moreover , ω is simple if and only if for any vectors

x1, . . . ,xk ∈ X,

P x1 ∧ P x2 ∧ · · · ∧ P xk = ω(x1 ∧ x2 ∧ · · · ∧ xk) w1 ∧ w2 ∧ · · · ∧ wk.

The proof of the lemma is left as an exercise to the reader.

Proof of Theorem 4.13. Assume that φ is a norm in Λn−1X and let W ⊂ X

be an (n−1)-dimensional subspace. Choose a basis of W , w1, . . . ,wn−1, such

that φ(w1 ∧ w2 ∧ · · · ∧ wn−1) = 1 and consider the support hyperplane of the

unit sphere of (Λn−1X,φ) at the point w1 ∧w2 ∧ · · ·∧wn−1. This hyperplane is


given by an equation of the form ω = 1, where ω is an (n−1)-form with constant

coefficients. In other words, ω ∈ Λn−1X∗.

We claim that the linear projection

Px :=n−1∑

i=1

(−1)iω(x ∧ w1 ∧ · · · ∧ wi ∧ · · · ∧ wn−1)wi

is φ-decreasing. Indeed, since ω is an (n−1)-form on an n-dimensional space, it

is simple. Using the second part of Lemma 4.14, we have, for any (n−1)-vector

a := x1 ∧ x2 ∧ · · · ∧ xn−1,

φ(P x1 ∧P x2 ∧ · · · ∧P xn−1) = |ω(a)|φ(w1 ∧w2 ∧ · · · ∧wn−1) = |ω(a)| ≤ φ(a).

To prove the converse, we note that the existence of a φ-decreasing linear

projection onto any given hyperplane implies that the (n−1)-volume of the facet

of any closed (n−1)-dimensional polyhedron is less than or equal to the sum of

the areas of the remaining facets. The argument is quite simple: if the closed

polyhedron has facets F0, F1, . . . , Fm, we use the φ-decreasing projection P to

project the whole polyhedron onto the hyperplane containing F0. Note that

P (F1)∪ · · · ∪ P (Fm) contains P (F0) = F0 and, therefore, the sum of the (n−1)-

volumes of the P (Fi), 1 ≤ i ≤ m, is greater than or equal to the (n−1)-volume

of F0. Since P is φ-decreasing, this gives us that the sum of the (n−1)-volumes

of the Fi, 1 ≤ i ≤ m, is greater than or equal to the (n−1)-volume of F0.

We now state the fourth and last of our characterizations of norms on the space

of (n−1)-vectors in an n-dimensional normed space.



• φ is a norm;

• If K ⊂ K ′ are two nested convex bodies in X, then the area of ∂K is less than

or equal to the area of ∂K ′ with equality if and only if K equals K ′.

The proof of this theorem is a simple consequence of the relation between norms

in the space of (n−1)-vectors and the theory of mixed volumes that will be

developed in Section 6.

At the beginning of this section we mentioned that for any n-dimensional

normed space X the Busemann, Holmes–Thompson, and mass∗ (n−1)-volume

densities on X are norms in Λn−1X. For the Busemann definition this is a

celebrated theorem of Busemann [1949a]. For the mass∗ definition this result is

due to Benson [1962]. We shall follow [Gromov 1983] and give a proof of a much

stronger result later in this section. For the Holmes–Thompson definition, the

result — under a different formulation — goes back to Minkowski.

Theorem 4.16 (Minkowski). The Holmes–Thompson (n−1)-volume density

of an n-dimensional normed space X is itself a norm in Λn−1X.


In order to prove the convexity of the Holmes–Thompson (n−1)-volume density,

we shall first give an integral representation for it. This representation depends,

in turn, on two classical constructions: the Gauss map and the surface-area

measure. Our approach follows [Fernandes 2002].

Let X be an n-dimensional vector space and let φ be an (n−1)-volume density

on X. If N ⊂ X is an oriented hypersurface and n ∈ N , we define Gφ(n) as

the unique (n−1)-vector in Λn−1TnN ⊂ Λn−1X that is positively oriented and

satisfies φ(Gφ(n)) = 1. Notice that when N is a strictly convex hypersurface,

the Gauss map

Gφ : N −→ Σ := a ∈ Λn−1X : φ(a) = 1

is a diffeomorphism. In this case, we define the surface-area measure of N as the

(n−1)-volume density dSN := G−1∗φ φ on Σ.

Lemma 4.17. Let π : X → Y be a surjective linear map between an n-

dimensional vector space X and an (n−1)-dimensional vector space Y , and let

φ be an (n−1)-volume density on X with unit sphere Σ ⊂ Λn−1X. If N ⊂ X is

a smooth, strictly convex hypersurface and ω is a volume form on Y , then

∫

π(N)

|ω| =1

2

∫

a∈Σ

|π∗ω(a)| dSN .

Proof. By the definition of the Gauss map, if n ∈ N and x1∧x2∧· · ·∧xn−1 ∈Λn−1TnN ,

x1 ∧ x2 ∧ · · · ∧ xn−1 = φ(x1 ∧ x2 ∧ · · · ∧ xn−1)Gφ(n).

Therefore, π∗|ω|(x1 ∧ x2 ∧ · · · ∧ xn−1) = |π∗ω(Gφ(n))|φ(x1 ∧ x2 ∧ · · · ∧ xn−1).

Then

∫

π(N)

|ω| =1

2

∫

N

π∗|ω| =1

2

∫

n∈N

|π∗ω(Gφ(n))|φ

=1

2

∫

Σ

G−1∗φ |π∗ω(Gφ(n))|φ =

1

2

∫

a∈Σ

|π∗ω(a)|G−1∗φ φ

=1

2

∫

a∈Σ

|π∗ω(a)| dSN .

Proof of Theorem 4.16. By a standard approximation argument, it suf-

fices to consider the case where the dual unit sphere ∂B∗ ⊂ X∗ is smooth and

strictly convex. Applying the previous lemma to the surface N = ∂B∗ and to

an arbitrary (n−1)-volume density on X∗, we have

µht(a) = ε−1n−1

∫

π(B∗)

|a| = ε−1n−1

∫

ξ∈Σ

|a(ξ)| dS∂B∗ .


Since the surface-area measure dS∂B∗ is nonnegative,

µht(a + b) = ε−1n−1

∫

ξ∈Σ

|a(ξ) + b(ξ)| dS∂B∗

≤ ε−1n−1

∫

ξ∈Σ

|a(ξ)| dS∂B∗ + ε−1n−1

∫

ξ∈Σ

|b(ξ)| dS∂B∗

= µht(a) + µht(b).

Exercise 4.18. Let X be an n-dimensional vector space and let φ be an (n−1)-

volume density on X. Show that if φ is a norm, then compact hypersurfaces

cannot by minimal.

4.3. Convexity properties of k-volume densities. We now pass to the

more delicate subject that Busemann, Ewald, and Shephard studied extensively

under the heading of convexity on Grassmannians. Most of what follows can be

found in their papers, “Convex bodies and convexity on Grassmannian cones”

I–XI, but we have tried to make the language and proofs more accessible.

We shall see that there are several notions and degrees of convexity for k-

volume densities. These are closely related to the concept of ellipticity in geo-

metric measure theory and, historically, to the generalization of the Legendre

condition for variational problems.

Weakly convex k-densities. Let X be an n-dimensional vector space and let

ΛksX, 1 ≤ k ≤ n − 1, be the cone of simple k-vectors on X. If Y is a (k + 1)-

dimensional subspace of X, then the subspace ΛkY ⊂ ΛkX lies inside ΛksX. This

motivates a definition:

Definition 4.19. A k-volume density φ on an n-dimensional vector space X,

n > k, is said to be weakly convex if for any linear subspace Y of dimension

k + 1, the restriction of φ to the linear space ΛkY is a norm.

From the previous section, we know that the k-volume densities of any normed

space for the Busemann, Holmes–Thompson, or mass∗ definitions of volume are

weakly convex.

Exercise 4.20. Show that a k-volume density in a vector space X is weakly

convex if for every (k + 1)-dimensional simplex in X the area of any one facet is

less than or equal to the sum of the areas of the remaining facets.

Extendibly convex k-volume densities

Definition 4.21. A k-volume density φ on an n-dimensional vector space X,

n > k, is said to be extendibly convex if it is the restriction of a norm on ΛkX

to the cone of simple k-vectors in X.

Equivalently, φ is extendibly convex if and only if there is a support hyperplane

for the unit sphere

S := a ∈ ΛksX : φ(a) = 1


passing through any of its points.

Theorem 4.22. If φ is an extendibly convex k-volume density on a vector

space X, then any k-chain with real coefficients whose image is contained in a

k-dimensional flat is φ-minimizing .

The proof — by the method of calibrations — is nearly identical to the proof of

Theorem 4.10 and is left as an exercise for the reader. Notice that a corollary to

Theorem 4.22 is that if P ⊂ X is a closed k-dimensional polyhedron, the area

of any of its facets is less than or equal to the sum of the areas of the remaining

facets.

The problem of determining whether the Busemann k-volume densities are

extendibly convex was posed by Busemann in several of his papers as a major

problem in convexity. So far, there are no results in this direction.

Problem. Is the Busemann 2-volume density of a 4-dimensional normed space

extendibly convex?

In the case of the Holmes–Thompson definition, Busemann, Ewald, and Shep-

hard have given explicit examples of norms for which the k-volume densities,

1 < k < n − 1, are not extendibly convex (see [Busemann et al. 1963]). A

simpler example has been given recently by Burago and Ivanov:

Theorem 4.23 [Burago and Ivanov 2002]. Consider the norm ‖ · ‖ on R4 whose

dual unit ball in R4∗ is the convex hull of the curve

γ(t) := (sin t, cos t, sin 3t, cos 3t), 0 ≤ t ≤ 2π.

The Holmes–Thompson 2-volume density for (R4, ‖ · ‖) is not extendibly convex .

Despite these examples, in many important cases the Holmes–Thompson k-

volume densities are extendibly convex.

Theorem 4.24. The Holmes–Thompson k-volume densities of a hypermetric

normed space are extendibly convex .

In order to prove this result, we shall derive a formula for the Holmes–Thompson

k-volume densities of a Minkowski space in terms of the Fourier transform of its

norm. In a somewhat different guise, this formula was first obtained by W. Weil

[1979]. In the present form it was rediscovered by Alvarez and Fernandes in

[1999], where it was shown to follow from the Crofton formula for Minkowski

spaces.

Let φ be a smooth, even, homogeneous function of degree one on an n-

dimensional vector space X, let e1, . . . ,en be a basis of X, and let ξ1, . . . , ξn be

the dual basis in X∗. Using the basis e1, . . . ,en and its dual to identify both X

and X∗ with Rn, we can compute the (distributional) Fourier transform of φ,

φ(ξ) :=

∫

Rn

eiξ·xφ(x) dx.


The form φ dξ1 ∧ · · · ∧ dξn does not depend on the choice of basis in X. Up to a

constant factor, we define the form φ as the contraction of this n-form with the

Euler vector field, XE(ξ) = ξ, in X∗:

φ :=−1

4(2π)n−1φ dξ1 ∧ · · · ∧ dξncXE .

It is known (see [Hormander 1983, pages 167–168]) that φ is smooth on X∗ \ 0

and homogeneous of degree −n− 1; therefore φ is a smooth differential form on

X∗ \ 0 that is homogeneous of degree −1.

Denoting by φk the product form in the product space (X∗ \0)k, we have the

following result:

Theorem 4.25. Let (X,φ) be an n-dimensional Minkowski space. For any

simple k-vector a on X, 1 ≤ k < n, we have

µht(a) =1

εk

∫

(ξ1,...ξ

k)∈S∗k

|ξ1 ∧ · · · ∧ ξk · a|φk,

where S∗ is any closed hypersurface in X∗ \ 0 that is star-shaped with respect to

the origin.

Notice that this formula allows us to extend the definition of the Holmes–

Thompson k-volume density of any Minkowski space to all of ΛkX. It remains

to see when this extension is a norm.

Proof of Theorem 4.24. It is enough to prove convexity in the case the

hypermetric space (X,φ) is also a Minkowski space. This allows us to use the

integral representation given above. Since X is hypermetric, Theorem 2.5 tells

us that the form φ dξ1∧· · ·∧dξn is a volume form, and, therefore, the restriction

of φk to the manifold S∗k defines a nonnegative measure. Then for any two

k-vectors a and b we have

µht(a + b) =

∫

S∗k

|ξ1 ∧ · · · ∧ ξk · (a + b)| φk

≤∫

S∗k

|ξ1 ∧ · · · ∧ ξk · a|φk +

∫

S∗k

|ξ1 ∧ · · · ∧ ξk · b|φk

= µht(a) + µht(b).

Totally convex k-densities

Definition 4.26. A k-density φ on an n-dimensional vector space X, n > k, is

said to be totally convex if through every point of the unit sphere of ΛkX there

passes a supporting hyperplane of the form ξ = 1 with ξ a simple k-vector in

ΛkX∗.

Total convexity implies extendible convexity and, in turn, weak convexity. The

following result, stated in [Busemann 1961] gives an important characterization

of totally convex k-densities in terms of what Gromov [1983] calls the compressing

property.


Theorem 4.27. A k-density φ on an n-dimensional vector space X is totally

convex if and only if for every k-dimensional linear subspace there exists a φ-

decreasing linear projection onto that subspace.

The proof, using Lemma 4.14, is nearly identical to the proof of Theorem 4.13.

Of all the four volume definitions we have studied, mass∗ has by far the

strongest convexity property:

Theorem 4.28. The mass∗ k-volume densities of an n-dimensional normed

space X, 1 ≤ k ≤ n − 1, are totally convex .

Proof. By Theorem 4.27, it is enough to show that given any k-dimensional

subspace W , there exists a linear projection P : X →W that is mass∗-decreasing.

Choose a basis ξ1, . . . , ξk of W ∗ which satisfies two properties:

(1) It is short (i.e., |ξi(x)| ≤ 1 for all x ∈ B ∩ W );

(2) The integral of the volume density |ξ1 ∧ξ2 ∧ · · ·∧ξk| over B∩W is maximal

among all short bases.

Notice that for any basis w1, . . . ,wk of W , we have that

µm∗(w1 ∧ w2 ∧ · · · ∧ wk) = |ξ1 ∧ ξ2 ∧ · · · ∧ ξk (w1 ∧ w2 ∧ · · · ∧ wk)|,

and that if w1, . . . ,wk is dual to ξ1, . . . , ξk, then µm∗(w1 ∧ w2 ∧ · · · ∧wk) = 1.

By the Hahn–Banach theorem, there exist covectors ξ1, . . . , ξk ∈ X∗ such

that

(1) |ξi(x)| ≤ 1 for all x ∈ B and for all i, 1 ≤ i ≤ k;

(2) the restriction of ξi to W equals ξi for all i, 1 ≤ i ≤ k.

We may now define the projection P : X → W by the formula

P (x) :=k

∑

i=1

ξi(x)wi,

and show that it is µm∗-decreasing. Indeed, if a = v1 ∧ v2 ∧ · · · ∧ vk is a simple

k-vector in X,

P (v1) ∧ P (v2) ∧ · · · ∧ P (vk) = ξ1 ∧ ξ2 ∧ · · · ∧ ξk (a)w1 ∧ w2 ∧ · · · ∧ wk,

and therefore

µm∗(P (v1) ∧ P (v2) ∧ · · · ∧ P (vk)) = |ξ1 ∧ ξ2 ∧ · · · ∧ ξk (a)|.

Since the restriction of ξi, 1 ≤ i ≤ k, to 〈a〉 form a short basis of 〈a〉∗, we have

µm∗(P (v1) ∧ P (v2) ∧ · · · ∧ P (vk)) = |ξ1 ∧ ξ2 ∧ · · · ∧ ξk (a)| ≤ µm∗(a).

We end the section with an exercise and an open problem:

Exercise 4.29. Show that the sum of two totally convex 2-volume densities in

R4 is not necessarily totally convex. On the other had, show that the maximum

of two totally convex k-volume densities is a totally convex k-volume density.


Problem. For what (hypermetric) normed spaces are the Holmes–Thompson

k-volume densities totally convex?

5. Length and Area in Two-Dimensional Normed Spaces

Before going into the rich and beautiful theory of (hypersurface) area on finite-

dimensional normed spaces, we shall sharpen our intuition by carefully consid-

ering the case of two-dimensional normed spaces. This case is fundamentally

simpler because the notion of hypersurface area coincides with that of length

and is thus independent of our volume definition. Nevertheless, we shall see

that the theory of length and area on two-dimensional normed spaces is far from

trivial and provides a platform from which to jump to higher dimensions.

We start with two theorems that involve solely the notion of length:

Theorem 5.1 [Go lab 1932]. The perimeter of the unit circle of a two-dimen-

sional normed space is between six and eight . Moreover , the length is equal to

six if and only if the unit ball is an affine regular hexagon and is equal to eight

if and only if it is a parallelogram.

Full proofs can be found in [Schaffer 1967] and [Thompson 1996]. We stress that

the length of the unit circle is measured with the definition of length given by

the norm:

If γ : [a, b] → X is a continuous curve on the normed space (X, ‖ · ‖), the

length of γ is defined as the supremum of the quantities

n−1∑

i=0

‖γ(ti+1) − γ(ti)‖

taken over all partitions a = t0 < t1 < · · · < tn = b of the interval [a, b]. Notice

that if γ is differentiable, we can also compute its length by the integral

`(γ) =

∫ b

a

‖γ(t)‖ dt.

It is convenient to denote the length of a curve γ on the normed space X

with unit disc B by `B(γ). Note that `B(∂B) is an affine invariant of the convex

body B.

Theorem 5.2 [Schaffer 1973]. If B and D are unit balls of two norms in a

two-dimensional space X and if B∗ and D∗ are the dual balls in X∗ then

`D(∂B) = `B∗(∂D∗).

In particular ,

`B(∂B) = `B∗(∂B∗).

A complete proof is available in [Thompson 1996].

For those who like simply stated open problems, we pass on the following

question of Schaffer (private communication):


Problem. Given an arbitrary convex body B ⊂ R3 that is symmetric with

respect to the origin, does there always exist a plane Π passing through the origin

and for which `Π∩B(∂(Π ∩ B)) is less than or equal to 2π?

Next we discuss the relation between length and area on two-dimensional normed

spaces. The first important question that arises is the isoperimetric problem: Of

all convex bodies in a two-dimensional normed space X with a given perimeter

find those that enclose the largest area.

The solution of this problem passes through the representation of the length

as a mixed volume (in this case a mixed area). This permits the use of Brunn–

Minkowski theory to solve the isoperimetric problem and to also give further

properties of the length functional. The reader is referred to [Schneider 1993]

for a complete discussion of the theory, but our needs can be met in just a few

paragraphs.

Let X be an n-dimensional vector space and let λ be a Lebesgue measure on

X. If K and L are two subsets of X, the Minkowski sum of K and L is the set

K + L := x + y ∈ X : x ∈ K,y ∈ L.

If L is the unit ball of a norm in X, we may think of K + L as the set of all

points in X whose distance from K is less than or equal to one. In other words,

the tube of radius one about the set K.

The mixed volume V (K[n − 1], L) of two closed, bounded convex sets K and

L in X is defined as a “directional derivative” of the Lebesgue measure:

V (K[n − 1], L) =1

nlim

t→+0

λ(K + tL) − λ(K)

t.

In the two-dimensional, case V (K,L) := V (K[1], L) is linear and monotonic in

each variable. The key result in the solution of the isoperimetric problem in

normed spaces is the Minkowski mixed volume inequality:

V (K[n − 1], L) ≥ λ(K)n−1λ(L).

Moreover, if K and L are convex bodies, then equality holds if and only if K is

obtained from L by translation and dilation.

Back to the two-dimensional case, if we’re given a centered convex body B,

we may define the magnitude of a vector x in two different ways:

(1) Take B to be the unit ball of a norm ‖ · ‖ on X and set the magnitude of x

to be ‖x‖.

(2) Let [x] ⊂ X denote the line segment from the origin to x and define the

magnitude of x as V ([x], B).

Exercise 5.3. Show that for any convex body B that is symmetric with respect

to the origin, the map x 7→ V ([x], B) is a norm, but that in general its unit disc

is different from B.


The first step in solving the isoperimetric problem in a normed space X is to

find a centrally symmetric convex body I such that

‖x‖ = V ([x], I), for all x ∈ X.

Of course, I will also depend on the choice of Lebesgue measure λ used to define

the mixed volume. However, given a volume definition the body I will be defined

intrinsically in terms of the norm.

The construction of I is extremely simple: Let B be the unit ball of X and

let Ω be the volume form on X that satisfies Ω(x∧y) = λ(x∧y) for all positive

bases x,y of X (we are forced to take an orientation of X at this point, but the

result will not depend on the choice). If

iΩ : X −→ X∗

is defined by iΩ(v)(w) := Ω(v ∧ w), the set I is given by (iΩB)∗.

Summarizing:

Proposition 5.4. Let X be a two-dimensional normed space with unit ball B

and volume form Ω. If I denotes the body (iΩB)∗ , then

‖x‖ = V ([x], I)


The proof will be postponed to the next section where we will treat the n-

dimensional version of the proposition.

Notice that if the orientation of X is changed, the form Ω changes sign, but

the symmetry of the unit disc B implies that the body I stays the same.

Exercise 5.5. Show that if K is a convex body in X, its perimeter equals

2V (K, I) and that, in particular, the perimeter of I is twice its area. Hint: Try

first with bodies whose boundaries are polygons and use the previous proposition.

The representation of length as a mixed volume gives an easy proof of the fol-

lowing monotonicity property of length in two-dimensional normed spaces.

Proposition 5.6. If K1 ⊂ K2 are nested convex bodies in a two-dimensional

normed space, then `(∂K1) ≤ `(∂K2).

The proof is left as a simple exercise to the reader. The following related exercise

is, perhaps, somewhat harder.

Exercise 5.7. Show that a Finsler metric on the plane satisfies the monotonicity

property in the previous proposition if and only if its geodesics are straight lines.

Theorem 5.8. Let X be a two-dimensional normed space with unit disc B and

area form Ω ∈ Λ2X∗. Of all convex bodies in X with a given perimeter the one

that encloses the largest area is, up to translations, a dilate of I := (iΩB)∗.


Proof. Let K ⊂ X be a convex body and let

`B(∂K) = 2V (K, I)

be its perimeter. By Minkowski’s mixed volume inequality, we have

`B(∂K)2

4= V (K, I)2 ≥ λ(K)λ(I)

with equality if an only if K and I are homothetic. Thus, the area enclosed by

K is maximal for a given perimeter if and only if K is a dilate of I.

Definition 5.9. Let Y 7→ (Λ2Y, µY ) be a volume definition for two-dimensional

normed spaces. If X is a two-dimensional normed space with unit ball B, the

isoperimetrix of X corresponding to the volume definition µ is the body IX :=(

iΩX(B)

)∗, where ΩX is a 2-form on X satisfying |ΩX | = µX .

We shall denote the isoperimetrices of a two-dimensional normed space X with

respect to the Busemann, Holmes–Thompson, mass, and mass∗ volume defini-

tions by IbX , I

htX , I

mX , and I

m∗X .

If T : X → X is an invertible linear transformation, the isoperimetrix, with

respect to any volume definition, of the norm with unit ball T (BX) is T (IX).

Exercise 5.10. If X is a two-dimensional normed space with unit ball B and

if µ is a particular choice of volume definition, then

`B(∂IX) = 2µX(IX) and µX(IX) = µ∗X(B∗).

Using this exercise, we can give sharp estimates on the area and perimeter of

the isoperimetrix of a two-dimensional normed space for Busemann, Holmes–

Thompson, and mass∗ volume definitions.

Indeed, it follows trivially from the exercise that µbX(Ib

X) = vp(BX)/π and

that µhtX (Iht

X ) = π. Using the Mahler and Blaschke–Santalo inequalities, we have

8

π≤ µb

X(IbX) ≤ π.

The fact that µm∗X (Im∗

X ) ≤ π with equality if and only if X is Euclidean is

equivalent to the inequality µm∗ ≥ µht for two-dimensional normed spaces.

Exercise 5.11. Find the sharp lower bound for µm∗X (Im∗

X ).

It is interesting to note that the Blaschke–Santalo inequality implies that

µbX(BX) ≥ µb

X(IbX) and µht

X (BX) ≤ µhtX (Iht

X ),

with equality in both cases if and only if B is an ellipse. Of course this implies

that for both the Busemann and Holmes–Thompson definitions BX = IX if

and only if X is Euclidean. Notice that whether a unit disc is equal to its

isoperimetrix depends on the volume definition we are using. However, whether

the unit disc is a dilate of its isoperimetrix does not depend on such a choice.


Definition 5.12. Let X be a two-dimensional normed space. If BX is a dilate

of IX for one (and, therefore, any) volume definition, the unit circle, ∂BX , is

said to be a Radon curve.

For comparison with the higher-dimensional case we summarize the properties

of the map I that sends a unit disc BX to IX . This maps sends convex bodies

to convex bodies; it is a bijection; it commutes with linear maps in the sense

that T (BX) is sent to T (IX) for all invertible linear maps T ; it maps polygons

to polygons, smooth bodies to strictly convex bodies and strictly convex bodies

to smooth bodies; and the only fixed points for the µb and µht normalizations

are ellipses.

A good, very elementary account of the construction of the isoperimetrix from

first principles and its relationship to physics and symplectic geometry (the ball

is used for measuring position and the isoperimetrix for measuring velocity) is

given by Wallen [1995].

Finally, we explore the relationship between the perimeter and area of the

unit ball. The motivation is that `(∂I) = 2µ(IB) and that in the Euclidean case

this holds for the ball.

Theorem 5.13. If X is a two-dimensional normed space with unit ball B then

2µm(B) ≤ `(∂B) ≤ 2µm∗(B)

with equality on the left if and only if ∂B is a Radon curve and on the right if

and only if ∂B is an equiframed curve.

For the definition of equiframed curves and a proof of the theorem we refer the

reader to [Martini et al. 2001] where the history of this result is also discussed.

Exercise 5.14. Use this result and properties of IX to show that BX = ImX if

and only if ∂BX is a Radon curve; and that BX = Im∗X if and only if ∂BX is

equiframed.

There is a further recent result in this direction.

Theorem 5.15 [Moustafaev]. If X is a two-dimensional normed space, then

2µhtX (BX) ≤ `(∂BX),

with equality if and only if X is Euclidean.

Proof. By definition of the isoperimetrix and Minkowski’s mixed volume in-

equality, we have

`(BX)2 = 4V (BX , IhtX ) ≥ 4µht

X (BX)µhtX (Iht

X ).

Using that µhtX (Iht

X ) = π and that µhtX (BX)/π ≤ 1, we have

`(BX)2 ≥ 4πµhtX (BX) ≥ 4πµht

X (BX)µht

X (BX)

π= 4µht

X (BX)2.


Exercise 5.16. If X is a two-dimensional normed space, show that

2 ≤ µmX(BX) ≤ π,

3 ≤ µm∗X (BX) ≤ 4,

and (using inequalities from Section 3)

8/π ≤ µhtX (BX) ≤ π.

Give the equality cases.

6. Area on Finite-Dimensional Normed Spaces

In Section 4, we saw that the Busemann, Holmes–Thompson, and mass∗ vol-

ume definitions induce k-volume densities that are weakly convex. In the special

case where the dimension of the normed space X is n = k + 1, then the (n−1)-

volume densities are norms on the space Λn−1X.

It follows from the properties of the volume definitions that, in all three cases,

the map that assigns to the normed space X the normed space Λn−1X has the

following properties:

(1) If T : X → Y is a short linear map between normed spaces X and Y , then

the induced map T∗ : Λn−1X → Λn−1Y is also short.

(2) The map X 7→ Λn−1X is continuous with respect to the topology induced

by the Banach–Mazur distance.

(3) If X is a Euclidean space, then the (n−1)-volume density is the standard

Euclidean area on X.

(4) If the dimension of X is two, the map X 7→ Λ1X is the identity.

Notice that property (1) states that for the Busemann, Holmes–Thompson, and

mass∗ definitions, the map that takes the normed space X to the normed space

Λn−1X is a covariant functor in the category N of finite-dimensional normed

spaces.

Definition 6.1. A definition of area on normed spaces assigns to every n-

dimensional, n ≥ 2, normed space X a normed space (Λn−1X,σX) in such a way

that properties (1)–(4) above are satisfied.

For simplicity, we shall speak of the Busemann, Holmes–Thompson, and mass∗definitions of area to refer to the definitions of area induced, respectively, by the

Busemann, Holmes–Thompson, and mass∗ volume definitions.

Definitions of area in normed spaces are related to important constructions in

convex geometry such as intersection bodies, projection bodies, and Wulff shapes.

However, let us start by posing a few natural questions that arise whenever we

have a definition of area. The answer to some of these questions, once specialized

to the Busemann and Holmes–Thompson definitions, are deep results in the


theory of convex bodies. Other questions are long-standing open problems, and

yet others seem to be new.

Given a definition of area X 7→ (Λn−1X,σX) on normed spaces, we may

ask: Is the map X 7→ (Λn−1X,σX) injective? What is its range? Does it

send crystalline norms to crystalline norms? Does it send Minkowski spaces to

Minkowski spaces? In what numeric range is the area of the unit sphere of an

n-dimensional normed space?

Other problems arise when we consider the relationship between length, area,

and volume, but, for now, let us concentrate on the questions we have just posed.

6.1. Injectivity and range of the area definition Let us start the study of

the injectivity and range of the Busemann definition of area by describing the

unit ball of the (n−1)-volume density in terms of a well-known construction in

convex geometry.

Busemann area and intersection bodies. Consider Rn with its Euclidean struc-

ture and its unit sphere Sn−1. If K ⊂ Rn is a star-shaped body containing

the origin, the intersection body of K, IK, is defined by the following simple

construction: if x ∈ Rn is a unit vector, let A(K ∩ x⊥) denote the area of the

intersection of K with the hyperplane perpendicular to x, and let IK be the

star-shaped body enclosed by the surface

x/A(K ∩ x⊥) ∈ Rn : x ∈ Sn−1.

A celebrated theorem of Busemann, which is equivalent to the weak convexity

of the Busemann volume definition, states that if K is a centered convex body,

then IK is also a centered convex body.

Let X be an n-dimensional normed space. Choose a basis of X and use it to

identify X with Rn. Take the Euclidean structure in R

n for which the basis is

orthonormal and use the resulting Euclidean structure to identify the spaces X ∗

and Λn−1X, as well as to define the unit sphere Sn−1 in Rn.

Exercise 6.2. Show that with all these identifications, the convex body x ∈R

n : σbX(x) ≤ 1 is εn−1 times the intersection body of BX ; (here σb

X is the

norm induced on X∗ by the norm on Λn−1X).

Notice that we can now write the question of whether the Busemann definition

of area is injective in the following classical form: Is a centered convex body

determined uniquely by the area of its intersections with hyperplanes passing

through the origin? The answer is affirmative (see [Lutwak 1988] and [Gardner

1995]), and so we have the following result:

Theorem 6.3. The Busemann area definition is injective.

Determining the range of the Busemann area definition is somewhat trickier.

Thanks to the efforts of R. Gardner, G. Zhang, and others in the solution of the

first of the Busemann–Petty problems, it is known (see [Gardner 1994], [Gard-

ner et al. 1999], and [Zhang 1999] and the references therein) that in dimensions


two, three, and four every convex body symmetric with respect to the origin

is the intersection body of some star-shaped body. It is not clear at this point

whether those bodies that are intersection bodies of centered convex bodies can

be characterized effectively. For dimensions greater than four, not every centered

convex body is an intersection body ([Gardner et al. 1999]). For further infor-

mation about intersection bodies see, for example, [Gardner 1995] and [Lutwak

1988].

Examples in [Thompson 1996] show that the Busemann area definition does

not take crystalline norms to crystalline norms. We don’t know whether it takes

Minkowski norms to Minkowski norms.

Let us now pass to the Holmes–Thompson definition.

Holmes–Thompson area and projection bodies. Consider Rn with its Euclidean

structure and its unit sphere Sn−1. If K ⊂ Rn is a convex body, the projection

body of K, ΠK, is given by the following simple construction: if x ∈ Rn is a unit

vector, let A(K|x⊥) denote the area of the orthogonal projection of K onto the

hyperplane perpendicular to x, and let the polar of ΠK be the body enclosed

by the surface

A(K|x⊥)x ∈ Rn : x ∈ Sn−1.

As in the case of the Busemann definition of area, identifying a normed space

X with Rn allows us to write the unit ball for the (n−1)-volume density in terms

of this nonintrinsic construction.

Exercise 6.4. Show that by identifying a normed space X with Rn as in the

previous exercise, the convex body x ∈ Rn : σht

X (x) ≤ 1 is 1/εn−1 times the

polar of the projection body of B∗X .

The question of the injectivity of the Holmes–Thompson definition of area can

now be formulated in classical terms: Is a centered convex body determined

uniquely by the area of its orthogonal projections onto hyperplanes? The answer,

in the affirmative, follows from a celebrated result of Alexandrov [1933] (see also

[Gardner 1995]). We then have the following result:

Theorem 6.5. The Holmes–Thompson area definition is injective.

It is known, basically from the time of Minkowski, that a centered convex body

B is the projection body of another if and only if it is a zonoid (see [Gardner

1995]). By Theorem 2.12, this means that for any n-dimensional normed space

X the normed space (Λn−1X,σhtX ) is hypermetric.

Moreover, because of the integral formula for the Holmes–Thompson (n−1)-

volume density in terms of the surface area measure of the dual sphere given

in the proof of Theorem 4.16, the problem of reconstructing the norm from

the Holmes–Thompson (n−1)-volume density is precisely the famous Minkowski

problem: Reconstruct a convex body from the knowledge of its Gauss curvature

as a function of its unit normals. The next two theorems follow directly from


the work of Minkowski, Pogorelov, and Nirenberg (see [Pogorelov 1978] for a

detailed presentation).

Theorem 6.6. The range of the Holmes–Thompson area definition is the set of

hypermetric normed spaces.

Theorem 6.7. Let X be an n-dimensional vector space and let σ : Λn−1X →[0,∞) be a Minkowski norm. If (Λn−1X,σ) is hypermetric, then there exists a

unique Minkowski norm ‖ · ‖ on X such that σ is the Holmes–Thompson (n−1)-

volume density of the normed space (X, ‖ · ‖).

Another important feature of the Holmes–Thompson area is the following (for a

proof see [Thompson 1996]):

Theorem 6.8. The Holmes–Thompson area definition takes Minkowski spaces

to Minkowski spaces and crystalline norms to crystalline norms.

Mass* area and wedge bodies. Let B be a centered convex body in an n-

dimensional vector space X, and let Bn−1 be the (n−1)-fold product of B in

the n(n − 1)-dimensional space Xn−1. If Alt : Xn−1 → Λn−1X denotes the

(nonlinear) map

(x1, . . . ,xn−1) 7−→ x1 ∧ x2 ∧ · · · ∧ xn−1,

we define the wedge body of B, denoted by WB, as the convex hull of Alt(Bn−1)

in Λn−1X.

We remark that even if Bn−1 ⊂ Xn−1 is a centered convex body, Alt(Bn−1)

is not necessarily convex.

Theorem 6.9. The unit ball in Λn−1X for the mass∗ (n−1)-volume density of

a normed space X is the body (WB∗X)∗.

Proof. By Exercise 4.3, we have

σm∗X (a) = sup|ξ1 ∧ ξ2 ∧ · · · ∧ ξn−1 · a| : ξ1, . . . , ξn−1 ∈ B∗

X.

But this is just the supremum of |η ·a|, where η ∈ Alt(

(B∗X)n−1

)

. Therefore σm∗X

is the dual to the norm in Λn−1X∗ whose unit ball is WB∗X .

It is quite easy to do calculations for WB∗ in the case when the centered convex

body B is a simple object. The following statements are based on such calcula-

tions, the details of which are left as exercises (see also [Thompson 1999]).

Proposition 6.10. The mass∗ area definition is not injective.

Sketch of the proof.. All we must do is find two centered convex bodies B

and K such that WB∗ = WK∗, but B 6= K.

Let B be the cube with vertices at (±1,±1,±1). In this case, B∗ is the

octahedron with vertices (±1, 0, 0), (0,±1, 0), (0, 0,±1) and WB∗ = B∗.


Let K be the cuboctahedron with vertices (±1,±1, 0), (±1, 0,±1), (0,±1,±1).

The dual ball K∗ is the rhombic dodecahedron with vertices ±(1, 0, 0), ±(0, 1, 0),

±(0, 0, 1) and (± 12 ,± 1

2 ,± 12 ). A simple calculation shows that WB∗ = WK∗.

In fact, if L is any centered convex body that lies between the cube and the

cube-octahedron then WL∗ = WB∗.

While it seems unlikely that the wedge body of the unit ball in a Minkowski

space is the ball of a Minkowski space, it is not hard to show that the wedge

body of a polytope is a polytope. Then:

Proposition 6.11 [Thompson 1999]. The mass∗ area definition takes crys-

talline norms to crystalline norms.

The question of determining the range for the mass∗ area definition is completely

open. Is it possible that any centered convex body is a wedge body?

6.2. Area of the unit sphere. In this section we give the higher-dimensional

analogues (as far as we know them) of the theorems of Schaffer and Go lab dis-

cussed in Section 5.

The Holmes–Thompson definition was designed originally to yield a general-

ization of Schaffer’s result and we have the following theorem.

Theorem 6.12 [Holmes and Thompson 1979]. If B and K are the unit balls of

two norms ‖ · ‖B and ‖ · ‖K in the vector space X, the Holmes–Thompson area

of ∂K in the normed space (X, ‖ · ‖B) equals the Holmes–Thompson area of ∂B∗

in the normed space (X∗, ‖ · ‖∗K).

Notice that in particular, the Holmes–Thompson area of the unit sphere of a

normed space equals the Holmes–Thompson area of the unit sphere of its dual.

Simple calculations show that neither the Busemann, the mass∗, nor the mass

definition have this property. In fact, Daniel Hug (private communication) has

shown that Theorem 6.12 characterizes the Holmes–Thompson definition. How-

ever, the following question remains open.

Problem [Thompson 1996]. Is the Holmes–Thompson definition of volume

characterized by the fact that the area of the unit sphere of a normed space

equals the area of the unit sphere of its dual?

The first result extending Go lab’s theorem to higher dimension is the following

sharp upper bound for the Busemann area of a unit sphere.

Theorem 6.13 [Busemann and Petty 1956]. The Busemann area of the unit

sphere of an n-dimensional normed space is at most 2nεn−1 with equality if and

only if B is a parallelotope.

For n ≥ 3 no sharp lower bound for the Busemann area of the unit sphere

of an n-dimensional normed space has been proved. It is conjectured that the

minimum is nεn attained by the Euclidean ball. However, when n = 3 it is also

attained by the rhombic dodecahedron.


Since µb ≥ µht, an upper bound for the Busemann area is also an upper bound

for the Holmes–Thompson area.

Corollary 6.14. The Holmes–Thompson area of the unit sphere of an n-

dimensional normed space is less than 2nεn−1.

While the sharp upper bound for the Holmes–Thompson area of the unit sphere

in any dimension greater than two is not known, the sharp lower bound in

dimension three is given by the following unpublished result of Alvarez, Ivanov,

and Thompson:

Theorem 6.15. The Holmes–Thompson area of the unit sphere of a three-

dimensional normed space is at least 36/π. Moreover , equality holds if the unit

ball is a cuboctahedron or a rhombic dodecahedron.

Since µb ≥ µht and µm∗ ≥ µht, we have the following lower bound for the

Busemann and mass∗ areas of the unit sphere of a three-dimensional normed

space.

Corollary 6.16. The Busemann and mass∗ areas of unit sphere of a three-

dimensional normed space is greater than 36/π.

Although these bounds are not sharp, they are the best bounds known so far.

It is possible to use a variety of inequalities including the Petty projection

inequality (in the case of σht) and the Busemann intersection inequality (in the

case of σb) to give nonsharp lower bounds. The reader is referred to [Thompson

1996] for examples of what one can get.

6.3. Mixed volumes and the isoperimetrix. We now pass to questions

concerning the relationship between areas and volumes, and, in particular, to

the solution of the isoperimetric problem in finite-dimensional normed spaces.

The subject is classical and has been studied from different viewpoints by convex

geometers, geometric measure theorists, and crystallographers (see, for example,

[Busemann 1949b], [Taylor 1978], and [Ambrosio and Kirchheim 2000]). Never-

theless, being interested in a particular intrinsic viewpoint and relations to area

on normed and Finsler spaces that are not treated elsewhere, we shall give a

short account of the subject.

Let X be an n-dimensional vector space and let λ be a Lebesgue measure on

X. If I ⊂ X is a centered convex body, we can define an (n−1)-volume density

on X by the following construction: given n − 1 linearly independent vectors

x1, . . . ,xn−1 ∈ X, we denote the parallelotope they define by [x1, . . . ,xn−1] and

set

σI(x1 ∧ x2 ∧ · · · ∧ xn−1) :=1

nlim

t→+0

λ([x1, . . . ,xn−1] + tI) − λ([x1, . . . ,xn−1])

t.

It is easy to see that σI is well defined and that by changing λ for another

Lebesgue measure on X we simply multiply σI by a constant. Note also that


although the measure of [x1, . . . ,xn−1] is zero, we have included it in the formula

to stress its relationship with the n-dimensional mixed volume of two bodies,

V (K[n − 1], L) :=1

nlim

t→+0

λ(K + tL) − λ(K)

t.

With this definition, if K is a convex body in X,∫

∂K

σI = nV (K[n − 1], I).

Exercise 6.17. Show that the (n−1)-volume density σI constructed above is a

norm on Λn−1(X), and that∫

∂I

σI = nλ(I).

We would also like to reverse this construction: Starting from a norm σ :

Λn−1X → [0,∞) and a Lebesgue measure λ on X construct a convex body

I ⊂ X such that σ = σI . The construction is quite simple: Let Ω be a volume

form on X such that |Ω| = λ and consider the linear isomorphism

iΩ : Λn−1X −→ X∗

defined by iΩ(x1 ∧x2 ∧ · · · ∧xn−1)(x) = Ω(x1 ∧x2 ∧ · · · ∧xn−1 ∧x). The body

I is given by (iΩB)∗, where B ⊂ Λn−1X is the unit ball of σ.

In terms of mixed volumes, we have the following result:

Proposition 6.18. Let X be an n-dimensional vector space, let σ be a norm

on Λn−1X with unit ball B and let λ be a Lebesgue measure on X. Using the

notation above, if I := (iΩB)∗, we have∫

∂K

σ = nV (K[n − 1], I)

for all convex bodies K ⊂ X.

To prove the proposition, let us give a simpler, more visual relationship between

σ and I := (iΩB)∗ that is of independent interest. Given a nonzero (n−1)-vector

a ∈ Λn−1X, we shall say that a vector v ∈ X is normal to a with respect to I

if v ∈ ∂I, the hyperplane parallel to 〈a〉 and passing through v supports I, and

Ω(a ∧ v) > 0. When I is smooth and strictly convex the normal is unique, but

this is of no importance to what follows. Notice, and this is important, that v

is constructed in such a way that

Ω(a ∧ v) = sup|Ω(a ∧ x)| : x ∈ I.

Lemma 6.19. Let X be an n-dimensional vector space, let σ be a norm on

Λn−1X with unit ball B and let Ω ∈ ΛnX∗ be a volume form on X. If a is a

nonzero (n−1)-vector on X and v ∈ X is normal to a with respect to I := (iΩB)∗,

then

σ(a) = Ω(a ∧ v).


Proof. Let ‖ · ‖∗ denote the norm in X∗ whose unit ball is I∗ = iΩB. Trivially,

we have σ(a) = ‖iΩ(a)‖∗ for any a ∈ Λn−1X. Therefore,

σ(a) = sup|Ω(a ∧ x)| : x ∈ I = Ω(a ∧ v).

In other terms, if x1, . . . ,xn−1 are linearly independent vectors in X and v is

normal to x1 ∧ x2 ∧ · · · ∧ xn−1 with respect to I, then the volume of the n-

dimensional parallelotope [x1, . . . ,xn−1,v] is the area of the (n−1)-dimensional

parallelotope [x1, . . . ,xn−1].

Proof of Proposition 6.18. Let x1, . . . ,xn−1 be linearly independent vectors

in X and let [x1, . . . ,xn−1] denote the parallelotope spanned by them. Notice

that if v is normal to x1 ∧ x2 ∧ · · · ∧xn−1 with respect to I, then for any t > 0,

the union of the n-dimensional parallelotopes

[x1, . . . ,xn−1, tv] and [x1, . . . ,xn−1,−tv],

which we denote by P (t), is contained in the set [x1, . . . ,xn−1] + tI. More-

over, since up to terms of order 2 and higher in t the volumes of P (t) and

[x1, . . . ,xn−1] + tI are the same, we have

1

nlim

t→+0

λ([x1, . . . ,xn−1] + tI)

t=

1

nlim

t→+0

λ(P (t))

t= σ(x1 ∧ x2 ∧ · · · ∧ xn−1),

and this concludes the proof.

We are now ready to solve the isoperimetric problem for convex bodies:

Theorem 6.20. Let X be an n-dimensional vector space, let σ be a norm on

Λn−1X with unit ball B, and let Ω ∈ ΛnX∗ be a volume form on X. Of all

convex bodies in X with a given surface area the one that encloses the largest

volume is, up to translations, a dilate of I := (iΩB)∗.

Proof. Let K ⊂ X be a convex body and let∫

∂K

σ = nV (K[n − 1], I)

be its surface area. By Minkowski’s mixed volume inequality, we have(

∫

∂K

σ

)n

= nnV (K[n − 1], I)n ≥ nnλ(K)n−1λ(I)

with equality if an only if K and I are homothetic. Thus, the volume enclosed

by K is maximal for a given surface area if and only if K is a dilate of I.

We shall denote the isoperimetrices of a normed space X with respect to the

Busemann, Holmes–Thompson, and mass∗ definitions by IbX , I

htX , and I

m∗X , re-

spectively. In the case of the Busemann and Holmes–Thompson definitions,

the isoperimetrices can be given, nonintrinsically, in terms of intersection and

projection bodies.


Exercise 6.21. Using Exercises 6.2 and 6.4, and the construction of the iso-

perimetrix, show that

IbX =

εn−1

εnλ(BX)(IBX)∗ and I

htX =

εn

εn−1λ∗(B∗

X)−1ΠB∗X ,

where λ and λ∗ are, respectively, the Euclidean volumes on X and X∗ given by

their identification with Rn.

Exercise 6.22. Describe the isoperimetrix Im∗X B in terms of wedge bodies.

6.4. Geometry of the isoperimetrix. We now turn our attention to problems

relating the unit ball of a normed space and its isoperimetrix with respect to some

volume definition. Let us start with the deceptively simple problem of estimating

the volume of the isoperimetrix.

Identifying the normed space X with Rn as in Exercise 6.21, we see that the

Holmes–Thompson volume of IhtX is

µhtX (Iht

X ) = ε−1n

(

εn

εn−1

)n

λ∗(B∗X)−n+1λ(ΠB∗

X).

The statement that this quantity is greater than or equal to εn with equality if

and only if X is Euclidean is known as Petty’s conjectured projection inequal-

ity, and is one of the major open problems in the theory of affine geometric

inequalities.

Sharp lower bounds for µbX(Ib

X) and µm∗X (Im∗

X ) are also unknown, although as

observed in [Thompson 1996] the inequality µbX(Ib

X) ≥ εn for n ≥ 3 would easily

yield (exercise!) that the Busemann area of unit sphere of a normed space of

dimension n is at least nεn.

Another interesting affine invariant involving the isoperimetrix is the sym-

plectic volume of BX × I∗X in X × X∗. In the two-dimensional case this simply

yields the square of the area of the unit disc, but in higher dimension it is a

much more interesting invariant:

Exercise 6.23. Pick up either [Gardner 1995] or [Thompson 1996] and, using

Exercise 6.21, prove that the inequality

svol(BX × I∗X) ≤ εnµX(BX)

is true for the Busemann (resp. Holmes–Thompson) definition by showing that

it is equivalent to Busemann’s intersection inequality (resp. Petty’s projection

inequality).

It would be interesting to complete the picture by having a sharp upper bound

for svol(BX × (Im∗X )∗) in terms of µm∗

X(BX).

We finish the paper by considering some questions relating length, area, and

volume. In terms of the isoperimetrix they have very simple statements: Given

a volume definition, when is the isoperimetrix equal to the unit ball, when is it

a multiple of the ball, and when is it inside the ball?


These simple questions are really about the existence of a coarea formula or

inequality for the different definitions of volume on normed and Finsler spaces.

Many Riemannian and Euclidean results depend, or seem to depend, on the

simple fact that volume = base × height. To what extent is this true in normed

and Finsler spaces?

In order to relate the coarea formula and inequality with the geometry of the

isoperimetrix, let us first define the height of a parallelotope [x1, . . . ,xn−1,xn]

in a vector space X with respect to a centered convex body B ⊂ X by the

following construction: let ξ ∈ X∗ be a covector in ∂B∗ such that ξ(xi) = 0 for

all i between 1 and n − 1. The quantity |ξ(xn)|, which is independent of the

choice of ξ, will be called the height of [x1, . . . ,xn−1,xn] with respect to B.

By the construction of the isoperimetrix, we know that the volume of the

parallelotope [x1, . . . ,xn−1,xn] equals the area of its base, [x1, . . . ,xn−1], times

its height with respect to the isoperimetrix. Therefore, if the volume of every

parallelotope in a normed space equals the area of its base times its height with

respect to the unit ball, the ball equals the isoperimetrix. If the volume is greater

than the area of the base times the height with respect to the unit ball, then the

isoperimetrix is contained in the ball, and so on.

The first clear sign that the relationship between length, area and volume

may not go smoothly on normed and Finsler spaces is the following result of

Thompson:

Proposition 6.24 [Thompson 1996]. The isoperimetric of a normed space X

for the Holmes–Thompson definition is contained in the unit ball if and only if

the space is Euclidean. In which case, the ball and the isoperimetric are equal .

In other words, the coarea equality or inequality “volume ≥ base × height” for

the Holmes–Thompson definition is true only for Euclidean spaces.

Proof. If IhtX ⊂ BX , then B∗

X ⊂ (IhtX )∗ and, therefore,

svol(BX × (IhtX )∗) ≥ svol(BX × B∗

X) = εnµhtX (BX).

By Exercise 6.23, the only way this can happen is if X is Euclidean.

However, for the mass∗ definition the coarea inequality is always true:

Theorem 6.25 [Gromov 1983]. If X is a finite-dimensional normed space, then

Im∗X ⊂ BX .

Proof. We must show that if [v1, . . . ,vn] is a parallelotope,

µm∗(v1 ∧ v2 ∧ · · · ∧ vn) ≥ σm∗(v1 ∧ v2 ∧ · · · ∧ vn−1)|ξ(vn)|,

where ξ ∈ ∂B∗X and ξ(vi) = 0, 1 ≤ i ≤ n − 1.

Without loss of generality we may suppose that v1,v2, . . . vn−1 is an extremal

basis in the subspace V ⊂ X they span, i.e. each vector vi is a point of contact

between BX ∩ V and a minimal circumscribing parallelotope for B ∩ V . Let u


be such that ‖u‖ = ξ(u) = 1 and set vn = αu+x where x ∈ V . The right hand

side of the above inequality is |α|.Let ξ1, ξ2, . . . , ξn−1 be the dual basis to the vi’s in V and extend these to the

whole of X by setting ξi(u) = 0. Then ξ1, ξ2, . . . , ξn−1, ξ are all of norm 1 and

form the dual basis to v1,v2, . . . ,vn−1,u. Now

µm∗X (v1 ∧ v2 ∧ · · · ∧ vn) = |α|µm∗

X (v1 ∧ v2 ∧ · · · ∧ vn−1 ∧ u)

= |α|(µmX∗(ξ1 ∧ ξ2 ∧ · · · ∧ ξn−1 ∧ ξ))−1

≥ |α|(‖ξ‖∏

‖ξi‖)−1 = |α|.

The inequality comes from the definition of mass.

As we have said, the problem of determining for what normed spaces metric

balls are solutions to the isoperimetric problem, i.e. when is the isoperimetrix

a multiple of the unit ball, is completely open for all three definitions of volume

in dimensions greater than two.

Acknowledgments

It is a pleasure to acknowledge the hospitality and wonderful working condi-

tions of the Research in Pairs program at the Mathematisches Forschungsinstitut

Oberwolfach, where the authors did a substantial part of the research going into

this paper. Alvarez also thanks the MAPA Institute at the Universite Catholique

de Louvain for its hospitality and technical support in the final stages of this

work. Thompson is grateful to NSERC for its support through grant #A-4066.

Many of the ideas in this paper concerning the abstract framework of the sub-

ject were first developed in the Geometry Seminar at the Universite Catholique

de Louvain and were greatly influenced by discussions with Emmanuel Fernan-

des and Gautier Berck. The thesis [Fernandes 2002] was for us a stepping-stone

for many of the subjects developed in this work.

References

[Alexandrov 1933] A. D. Aleksandrov, “A theorem on convex polyhedra” (Russian),Trudy Fiz.-Mat. Inst. Steklov. 4 (1933), 87.

[Alvarez 2002] J. C. Alvarez Paiva, “Dual mixed volumes and isosystolic inequalities”,preprint, 2002.

[Alvarez and Fernandes 1998] J. C. Alvarez Paiva and E. Fernandes, “Crofton formulasin projective Finsler spaces”, Electronic Research Announcements of the Amer.

Math. Soc. 4 (1998), 91–100.

[Alvarez and Fernandes 1999] J. C. Alvarez Paiva and E. Fernandes, “Fourier trans-forms and the Holmes–Thompson volume of Finsler manifolds”, Int. Math. Res.

Notices 19 (1999), 1032–104.

[Ambrosio and Kirchheim 2000] L. Ambrosio and B. Kirchheim, “Currents in metricspaces”, Acta Math. 185 (2000), 1–80.


[Benson 1962] R. V. Benson, “The geometry of affine areas”, Ph.D. Thesis, Universityof Southern California, Los Angeles, (1962). (University Microfilms Inc., Ann Arbor,Michigan 62-6037).

[Bolker 1969] E. D. Bolker, “A class of convex bodies”, Trans. Amer. Math. Soc. 145

(1969), 323–345.

[Bourgain and Lindenstrauss 1988] J. Bourgain and J. Lindenstrauss, “Projectionbodies”, pp. 250–270 in Geometric aspects of functional analysis (Tel Aviv, 1986/87),edited by J. Lindenstrauss and V. D. Milman, Lecture Notes in Math. 1317,Springer, Berlin, 1988.

[Burago and Ivanov 2002] D. Burago and S. Ivanov, “On asymptotic volume of Finslertori, minimal surfaces in normed spaces, and symplectic filling volume”, Ann. of

Math. (2) 156:3 (2002), 891–914.

[Busemann 1947] H. Busemann, “Intrinsic area”, Ann. of Math. (2) 48 (1947), 234–267.

[Busemann 1949a] H. Busemann, “A theorem on convex bodies of the Brunn–Min-kowski type”, Proc. Nat. Acad. Sci. USA 35 (1949), 27–31.

[Busemann 1949b] H. Busemann, “The isoperimetric problem for Minkowski area”,Amer. J. Math. 71 (1949), 743–762.

[Busemann 1961] H. Busemann, “Convexity on Grassmann manifolds”, Enseign. Math.

(2) 7 (1961), 139–152.

[Busemann 1969] H. Busemann, “Convex bodies and convexity on Grassmann cones:XI”, Math. Scand. 24 (1969), 93–101.

[Busemann and Petty 1956] H. Busemann and C. M. Petty, “Problems on convexbodies”, Math. Scand. 4 (1956), 88–94.

[Busemann et al. 1963] H. Busemann, G. Ewald, and G. C. Shephard, “Convex bodiesand convexity on Grassmann cones: I–IV”, Math. Ann. 151 (1963), 1–41.

[Busemann et al. 1962] H. Busemann, G. Ewald, and G. C. Shephard, “Convex bodiesand convexity on Grassmann cones: V”, Arch. Math. 13 (1962), 512–526.

[Busemann and Sphephard 1965] H. Busemann and G. C. Shephard, “Convex bodiesand convexity on Grassmann cones: X”, Ann. Mat. Pura Appl. 70 (1965), 271–294.

[Cohn 1980] D. L. Cohn, Measure theory, Birkhauser, Basel, 1980.

[Duran 1998] C. E. Duran, “A volume comparison theorem for Finsler manifolds”,Proc. Amer. Math. Soc. 126 (1998), 3079–3082.

[Ewald 1964a] G. Ewald, “Convex bodies and convexity on Grassmann cones: VII”,Abh. Math. Sem. Univ. Hamburg 27 (1964), 167–170.

[Ewald 1964b] G. Ewald, “Convex bodies and convexity on Grassmann cones: IX”,Math. Ann. 157 (1964), 219–230.

[Fernandes 2002] E. Fernandes, “Double fibrations: a modern approach to integral ge-ometry and Crofton formulas in projective Finsler spaces”, Ph.D. Thesis, UniversiteCatholique de Louvain, 2002.

[Gardner 1995] R. J. Gardner, Geometric Tomography, Encyclopedia of Mathematicsand its Applications 58, Cambridge University Press, Cambridge, 1995.

[Gardner 1994] R. J. Gardner, “A positive answer to the Busemann–Petty problem inthree dimensions”, Ann. of Math. (2) 140:2 (1994), 435–447.

[Gardner et al. 1999] R. J. Gardner, A. Koldobsky, and T. Schlumprecht, “An analyticsolution to the Busemann–Petty problem on sections of convex bodies”, Ann. of

Math. 149:2 (1999), 691–703.


[Go lab 1932] S. Go lab, “Quelques problemes metriques de la geometrie de Minkowski”,Trav. l’Acad. Mines Cracovie 6 (1932), 1–79.

[Goodey and Weil 1993] P. Goodey and W. Weil, “Zonoids and generalisations”,pp. 1297–1326 in Handbook of convex geometry, Vol. B, edited by P. M. Gruberet al., North-Holland, Amsterdam, 1993.

[Gromov 1967] M. Gromov, “On a geometric hypothesis of Banach” (Russian), Izv.

Akad. Nauk SSSR Ser. Mat. 31 (1967), 1105–1114.

[Gromov 1983] M. Gromov, “Filling Riemannian manifolds”, J. Diff. Geom. 18 (1983),1–147.

[Harvey and Lawson 1982] R. Harvey and H. B. Lawson, “Calibrated geometries”, Acta

Math. 148 (1982), 47–157.

[Holmes and Thompson 1979] R. D. Holmes and A. C. Thompson, “N -dimensionalarea and content in Minkowski spaces”, Pacific J. Math. 85 (1979), 77–110.

[Hormander 1994] L. Hormander, Notions of convexity, Progress in Mathematics 127,Birkhauser, Boston, 1994.

[Hormander 1983] L. Hormander, The Analysis of Linear Partial Differential Operators

I, Grundlehren der mathematischen Wissenschaften 256, Springer, Berlin, 1983.

[Ivanov 2002] S. Ivanov, “On two-dimensional minimal fillings”, St. Petersburg Math.

J. 13 (2002), 17–25.

[John 1948] F. John, “Extremum problems with inequalities as subsidiary conditions”,pp. 187–204 in Studies and essays presented to R. Courant on his 60th birthday,edited by K. O. Friedrichs, O. Neuegebauer, and J. J. Stoker, Interscience, NewYork, 1948.

[Lutwak 1988] E. Lutwak, “Intersection bodies and dual mixed volumes”, Adv. Math.

71 (1988), 232–261.

[Mahler 1939] K. Mahler, “Ein Minimalproblem fur konvexe Polygone”, Mathematica

(Zutphen) B7 (1939), 118–127.

[Martini et al. 2001] H. Martini, K. J. Swanepoel, and Weiß, G., “The geometry ofMinkowski spaces: a survey, part I”, Exposition. Math. 19 (2001), 97–142.

[McKinney 1974] J. R. McKinney, “On maximal simplices inscribed to a central convexset”, Mathematika 21 (1974), 38–44.

[Moustafaev] Z. Moustafaev, “The ratio of the length of the unit circle to the area ofthe unit disc in Minkowski planes”, to appear in Proc. Amer. Math. Soc.

[Nirenberg 1953] L. Nirenberg, “The Weyl and Minkowski problems in differentialgeometry in the large”, Comm. Pure Appl. Math. 6 (1953), 337–394.

[Pogorelov 1978] A. V. Pogorelov, The Minkowski multidimensional problem, ScriptaSeries in Mathematics, V. H. Winston, Washington (DC) and Halsted Press, NewYork, 1978.

[Reisner 1985] S. Reisner, “Random polytopes and the volume-product of symmetricconvex bodies”, Math. Scand. 57 (1985), 386–392.

[Reisner 1986] S. Reisner, “Zonoids with minimal volume product”, Math. Z. 192

(1986), 339–346.

[Schaffer 1967] J. J. Schaffer, “Inner diameter, perimeter, and girth of spheres”, Math.

Ann. 173 (1967), 59–79.

[Schaffer 1973] J. J. Schaffer, “The self-circumference of polar convex bodies”, Arch.

Math. 24 (1973), 87–90.


[Schneider 1993] R. Schneider, Convex bodies: the Brunn–Minkowski Theory, Encyclo-pedia of Math. and Its Appl. 44, Cambridge University Press, New York, 1993.

[Schneider 2001] R. Schneider, “On the Busemann area in Minkowski spaces”, Beitr.

Algebra Geom. 42 (2001), 263–273.

[Schneider 2001] R. Schneider, “Crofton formulas in hypermetric projective Finslerspaces”, Festschrift: Erich Lamprecht. Arch. Math. (Basel) 77:1 (2001), 85–97.

[Schneider 2002] R. Schneider, “On integral geometry in projective Finsler spaces”,Izv. Nats. Akad. Nauk Armenii Mat. 37 (2002) 34–51.

[Schneider and Weil 1983] R. Schneider and W. Weil, “Zonoids and related topics”,pp. 296–317 in Convexity and its applications, edited by P. M. Gruber and J. M.Wills, Birkhauser, Basel, 1983.

[Schneider and Wieacker 1997] R. Schneider and J. A. Wieacker, “Integral geometryin Minkowski spaces”, Adv.in Math. 129 (1997), 222–260.

[Shephard 1964a] G. C. Shephard, “Convex bodies and convexity on Grassmann cones:VI”, J. London Math. Soc. 39 (1964), 307–319.

[Shephard 1964b] G. C. Shephard, “Convex bodies and convexity on Grassmann cones:VIII”, J. London Math. Soc. 39 (1964), 417–423.

[Taylor 1978] J. Taylor, “Crystalline variational problems”, Bull. Amer. Math. Soc.

84:4 (1978), 568–588.

[Thompson 1996] A. C. Thompson, Minkowski Geometry, Encyclopedia of Mathemat-ics and its Applications 63, Cambridge University Press, New York, 1996.

[Thompson 1999] A. C. Thompson, “On Benson’s definition of area in Minkowskispace”, Canad. Math. Bull. 42:2 (1999) 237–247.

[Wallen 1995] L. J. Wallen, “Kepler, the taxicab metric and beyond: an isoperimetricprimer”, College J. Math. 26 (1995), 178–190.

[Weil 1979] W. Weil, “Centrally symmetric convex bodies and distributions II”, Israel

J. Math. 32 (1979), 173–182.

[Zhang 1999] G. Zhang, “A positive solution to the Busemann–Petty problem in R4”.

Ann. of Math. (2) 149:2 (1999), 535–543.

J. C. Alvarez PaivaDepartment of MathematicsPolytechnic UniversitySix MetroTech CenterBrooklyn, New York, 11201United States

jalvarez duke.poly.edu

A. C. ThompsonDepartment of Mathematics and StatisticsDalhousie UniversityHalifax, Nova ScotiaCanada B3H 3J5

[email protected]

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